Edited, memorised or added to reading list

on 09-Mar-2018 (Fri)

Do you want BuboFlash to help you learning these things? Click here to log in or create user.

#exponential-family
The exponential family of distributions provides a general framework for selecting a possible alternative parameterisation of the distribution, in terms of natural parameters, and for defining useful sample statistics, called the natural sufficient statistics of the family.

statusnot read reprioritisations
last reprioritisation on reading queue position [%]
started reading on finished reading on

Exponential family - Wikipedia
ions to consider. The concept of exponential families is credited to [1] E. J. G. Pitman, [2] G. Darmois, [3] and B. O. Koopman [4] in 1935–36. The term exponential class is sometimes used in place of "exponential family". [5] <span>The exponential family of distributions provides a general framework for selecting a possible alternative parameterisation of the distribution, in terms of natural parameters, and for defining useful sample statistics, called the natural sufficient statistics of the family. Contents [hide] 1 Definition 1.1 Examples of exponential family distributions 1.2 Scalar parameter 1.3 Factorization of the variables involved 1.4 Vector parameter 1.5 Vect




#inner-product-space #vector-space
A vector space (also called a linear space) is a collection of objects called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars.

statusnot read reprioritisations
last reprioritisation on reading queue position [%]
started reading on finished reading on

Vector space - Wikipedia
company, see Vector Space Systems. [imagelink] Vector addition and scalar multiplication: a vector v (blue) is added to another vector w (red, upper illustration). Below, w is stretched by a factor of 2, yielding the sum v + 2w. <span>A vector space (also called a linear space) is a collection of objects called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars. Scalars are often taken to be real numbers, but there are also vector spaces with scalar multiplication by complex numbers, rational numbers, or generally any field. The operations of v




Flashcard 1732731866380

Tags
#inner-product-space #vector-space
Question
A [...] is a collection vectors that can be added together and multiplied ("scaled") by numbers.
Answer
vector space


statusnot learnedmeasured difficulty37% [default]last interval [days]               
repetition number in this series0memorised on               scheduled repetition               
scheduled repetition interval               last repetition or drill

Parent (intermediate) annotation

Open it
A vector space (also called a linear space) is a collection of objects called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars.

Original toplevel document

Vector space - Wikipedia
company, see Vector Space Systems. [imagelink] Vector addition and scalar multiplication: a vector v (blue) is added to another vector w (red, upper illustration). Below, w is stretched by a factor of 2, yielding the sum v + 2w. <span>A vector space (also called a linear space) is a collection of objects called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars. Scalars are often taken to be real numbers, but there are also vector spaces with scalar multiplication by complex numbers, rational numbers, or generally any field. The operations of v







#inner-product-space #vector-space
Infinite-dimensional vector spaces arise naturally in mathematical analysis, as function spaces, whose vectors are functions.

These vector spaces are generally endowed with additional structure, which may be a topology, allowing the consideration of issues of proximity and continuity.

Among these topologies, those that are defined by a norm or inner product are more commonly used, as having a notion of distance between two vectors. This is particularly the case of Banach spaces and Hilbert spaces, which are fundamental in mathematical analysis.

statusnot read reprioritisations
last reprioritisation on reading queue position [%]
started reading on finished reading on

Vector space - Wikipedia
roperties, which in some cases can be visualized as arrows. Vector spaces are the subject of linear algebra and are well characterized by their dimension, which, roughly speaking, specifies the number of independent directions in the space. <span>Infinite-dimensional vector spaces arise naturally in mathematical analysis, as function spaces, whose vectors are functions. These vector spaces are generally endowed with additional structure, which may be a topology, allowing the consideration of issues of proximity and continuity. Among these topologies, those that are defined by a norm or inner product are more commonly used, as having a notion of distance between two vectors. This is particularly the case of Banach spaces and Hilbert spaces, which are fundamental in mathematical analysis. Historically, the first ideas leading to vector spaces can be traced back as far as the 17th century's analytic geometry, matrices, systems of linear equations, and Euclidean vectors.




#inner-product-space #vector-space
Infinite-dimensional vector spaces arise naturally in mathematical analysis, as function spaces, whose vectors are functions.

statusnot read reprioritisations
last reprioritisation on reading queue position [%]
started reading on finished reading on


Parent (intermediate) annotation

Open it
Infinite-dimensional vector spaces arise naturally in mathematical analysis, as function spaces, whose vectors are functions. These vector spaces are generally endowed with additional structure, which may be a topology, allowing the consideration of issues of proximity and continuity. Among these topologies, t

Original toplevel document

Vector space - Wikipedia
roperties, which in some cases can be visualized as arrows. Vector spaces are the subject of linear algebra and are well characterized by their dimension, which, roughly speaking, specifies the number of independent directions in the space. <span>Infinite-dimensional vector spaces arise naturally in mathematical analysis, as function spaces, whose vectors are functions. These vector spaces are generally endowed with additional structure, which may be a topology, allowing the consideration of issues of proximity and continuity. Among these topologies, those that are defined by a norm or inner product are more commonly used, as having a notion of distance between two vectors. This is particularly the case of Banach spaces and Hilbert spaces, which are fundamental in mathematical analysis. Historically, the first ideas leading to vector spaces can be traced back as far as the 17th century's analytic geometry, matrices, systems of linear equations, and Euclidean vectors.




Flashcard 1733076061452

Tags
#inner-product-space #vector-space
Question
[...] arise naturally in mathematical analysis, as function spaces, whose vectors are functions.
Answer
Infinite-dimensional vector spaces

Think multivariate Gaussian and Gaussian Process


statusnot learnedmeasured difficulty37% [default]last interval [days]               
repetition number in this series0memorised on               scheduled repetition               
scheduled repetition interval               last repetition or drill

Parent (intermediate) annotation

Open it
Infinite-dimensional vector spaces arise naturally in mathematical analysis, as function spaces, whose vectors are functions.

Original toplevel document

Vector space - Wikipedia
roperties, which in some cases can be visualized as arrows. Vector spaces are the subject of linear algebra and are well characterized by their dimension, which, roughly speaking, specifies the number of independent directions in the space. <span>Infinite-dimensional vector spaces arise naturally in mathematical analysis, as function spaces, whose vectors are functions. These vector spaces are generally endowed with additional structure, which may be a topology, allowing the consideration of issues of proximity and continuity. Among these topologies, those that are defined by a norm or inner product are more commonly used, as having a notion of distance between two vectors. This is particularly the case of Banach spaces and Hilbert spaces, which are fundamental in mathematical analysis. Historically, the first ideas leading to vector spaces can be traced back as far as the 17th century's analytic geometry, matrices, systems of linear equations, and Euclidean vectors.







Flashcard 1735821233420

Tags
#history #logic
Question
Johannes Gutenberg introduced new printing techniques in Europe around [...].
Answer
1440

You can't terrorise Aristotle!


statusnot learnedmeasured difficulty37% [default]last interval [days]               
repetition number in this series0memorised on               scheduled repetition               
scheduled repetition interval               last repetition or drill

Parent (intermediate) annotation

Open it
It is also not happenstance that the downfall of the disputational culture roughly coincided with the introduction of new printing techniques in Europe by Johannes Gutenberg, around 1440.

Original toplevel document

The rise and fall and rise of logic | Aeon Essays
ich is thoroughly disputational, with Meditations on First Philosophy (1641) by Descartes, a book argued through long paragraphs driven by the first-person singular. The nature of intellectual enquiry shifted with the downfall of disputation. <span>It is also not happenstance that the downfall of the disputational culture roughly coincided with the introduction of new printing techniques in Europe by Johannes Gutenberg, around 1440. Before that, books were a rare commodity, and education was conducted almost exclusively by means of oral contact between masters and pupils in the form of expository lectures in which







#stochastics
The term random function is also used to refer to a stochastic or random process,[25][26] because a stochastic process can also be interpreted as a random element in a function space.

statusnot read reprioritisations
last reprioritisation on reading queue position [%]
started reading on finished reading on

Stochastic process - Wikipedia
are considered the most important and central in the theory of stochastic processes, [1] [4] [23] and were discovered repeatedly and independently, both before and after Bachelier and Erlang, in different settings and countries. [21] [24] <span>The term random function is also used to refer to a stochastic or random process, [25] [26] because a stochastic process can also be interpreted as a random element in a function space. [27] [28] The terms stochastic process and random process are used interchangeably, often with no specific mathematical space for the set that indexes the random variables. [27] [29]




#stochastics
If the random variables are indexed by the Cartesian plane or some higher-dimensional Euclidean space, then the collection of random variables is usually called a random field instead.

statusnot read reprioritisations
last reprioritisation on reading queue position [%]
started reading on finished reading on

Stochastic process - Wikipedia
hangeably, often with no specific mathematical space for the set that indexes the random variables. [27] [29] But often these two terms are used when the random variables are indexed by the integers or an interval of the real line. [5] [29] <span>If the random variables are indexed by the Cartesian plane or some higher-dimensional Euclidean space, then the collection of random variables is usually called a random field instead. [5] [30] The values of a stochastic process are not always numbers and can be vectors or other mathematical objects. [5] [28] Based on their properties, stochastic processes can be d




#stochastics
the Wiener process or Brownian motion process,[a] used by Louis Bachelier to study price changes on the Paris Bourse

statusnot read reprioritisations
last reprioritisation on reading queue position [%]
started reading on finished reading on

Stochastic process - Wikipedia
arkets have motivated the extensive use of stochastic processes in finance. [16] [17] [18] Applications and the study of phenomena have in turn inspired the proposal of new stochastic processes. Examples of such stochastic processes include <span>the Wiener process or Brownian motion process, [a] used by Louis Bachelier to study price changes on the Paris Bourse, [21] and the Poisson process, used by A. K. Erlang to study the number of phone calls occurring in a certain period of time. [22] These two stochastic processes are considered the mo




#stochastics
Each random variable in the collection takes values from the same mathematical space known as the state space.

statusnot read reprioritisations
last reprioritisation on reading queue position [%]
started reading on finished reading on

Stochastic process - Wikipedia
element in the set. [4] [5] The set used to index the random variables is called the index set. Historically, the index set was some subset of the real line, such as the natural numbers, giving the index set the interpretation of time. [1] <span>Each random variable in the collection takes values from the same mathematical space known as the state space. This state space can be, for example, the integers, the real line or n {\displaystyle n} -dimensional Euclidean space. [1] [5] An increment i




#stochastics
The set used to index the random variables is called the index set.

statusnot read reprioritisations
last reprioritisation on reading queue position [%]
started reading on finished reading on

Stochastic process - Wikipedia
stochastic or random process can be defined as a collection of random variables that is indexed by some mathematical set, meaning that each random variable of the stochastic process is uniquely associated with an element in the set. [4] [5] <span>The set used to index the random variables is called the index set. Historically, the index set was some subset of the real line, such as the natural numbers, giving the index set the interpretation of time. [1] Each random variable in the collection t




#stochastics
A stochastic process can have many outcomes, due to its randomness, and a single outcome of a stochastic process is called, among other names, a sample function or realization

statusnot read reprioritisations
last reprioritisation on reading queue position [%]
started reading on finished reading on

Stochastic process - Wikipedia
r n {\displaystyle n} -dimensional Euclidean space. [1] [5] An increment is the amount that a stochastic process changes between two index values, often interpreted as two points in time. [48] [49] <span>A stochastic process can have many outcomes, due to its randomness, and a single outcome of a stochastic process is called, among other names, a sample function or realization. [28] [50] [imagelink] A single computer-simulated sample function or realization, among other terms, of a three-dimensional Wiener or Brownian motion process for time 0 ≤ t ≤ 2.




#stochastics
A stochastic process can be classified in different ways, for example, by
  1. its state space,
  2. its index set, or
  3. the dependence among the random variables.

statusnot read reprioritisations
last reprioritisation on reading queue position [%]
started reading on finished reading on

Stochastic process - Wikipedia
f a three-dimensional Wiener or Brownian motion process for time 0 ≤ t ≤ 2. The index set of this stochastic process is the non-negative numbers, while its state space is three-dimensional Euclidean space. Classifications[edit source] <span>A stochastic process can be classified in different ways, for example, by its state space, its index set, or the dependence among the random variables. One common way of classification is by the cardinality of the index set and the state space. [51] [52] [53] When interpreted as time, if the index set of a stochastic process has a fi




#stochastics
One of the simplest stochastic processes is the Bernoulli process,[60] which is a sequence of independent and identically distributed (iid) Bernoulli variables.

statusnot read reprioritisations
last reprioritisation on reading queue position [%]
started reading on finished reading on

Stochastic process - Wikipedia
} -dimensional vector process or n {\displaystyle n} -vector process. [51] [52] Examples of stochastic processes[edit source] Bernoulli process[edit source] Main article: Bernoulli process <span>One of the simplest stochastic processes is the Bernoulli process, [60] which is a sequence of independent and identically distributed (iid) random variables, where each random variable takes either the value one or zero, say one with probability p {\displaystyle p} and zero with probability 1 − p {\displaystyle 1-p} . This process can be likened to somebody flipping a coin, where the probability of obtaining a head is p {\displaystyle p} and its value is on




#stochastics
Random walks are stochastic processes that are usually defined as sums of iid random variables or random vectors in Euclidean space, so they are processes that change in discrete time.

statusnot read reprioritisations
last reprioritisation on reading queue position [%]
started reading on finished reading on

Stochastic process - Wikipedia
one, while the value of a tail is zero. [61] In other words, a Bernoulli process is a sequence of iid Bernoulli random variables, [62] where each coin flip is a Bernoulli trial. [63] Random walk[edit source] Main article: Random walk <span>Random walks are stochastic processes that are usually defined as sums of iid random variables or random vectors in Euclidean space, so they are processes that change in discrete time. [64] [65] [66] [67] [68] But some also use the term to refer to processes that change in continuous time, [69] particularly the Wiener process used in finance, which has led to some c




#stochastics
A classic example of a random walk is known as the simple random walk, which is a stochastic process in discrete time with the integers as the state space, and is based on a Bernoulli process, where each iid Bernoulli variable takes either the value positive one or negative one.

statusnot read reprioritisations
last reprioritisation on reading queue position [%]
started reading on finished reading on

Stochastic process - Wikipedia
ere are other various types of random walks, defined so their state spaces can be other mathematical objects, such as lattices and groups, and in general they are highly studied and have many applications in different disciplines. [69] [71] <span>A classic example of a random walk is known as the simple random walk, which is a stochastic process in discrete time with the integers as the state space, and is based on a Bernoulli process, where each iid Bernoulli variable takes either the value positive one or negative one. In other words, the simple random walk takes place on the integers, and its value increases by one with probability, say, p {\displaystyle p}




#stochastics
Playing a central role in the theory of probability, the Wiener process is often considered the most important and studied stochastic process, with connections to other stochastic processes.[1][2][3][78][79][80][81] Its index set and state space are the non-negative numbers and real numbers, respectively, so it has both continuous index set and states space.

statusnot read reprioritisations
last reprioritisation on reading queue position [%]
started reading on finished reading on

Stochastic process - Wikipedia
wnian motion due to its historical connection as a model for Brownian movement in liquids. [75] [76] [76] [77] [imagelink] Realizations of Wiener processes (or Brownian motion processes) with drift (blue) and without drift (red). <span>Playing a central role in the theory of probability, the Wiener process is often considered the most important and studied stochastic process, with connections to other stochastic processes. [1] [2] [3] [78] [79] [80] [81] Its index set and state space are the non-negative numbers and real numbers, respectively, so it has both continuous index set and states space. [82] But the process can be defined more generally so its state space can be n {\displaystyle n} -dimensional Euclidean space. [71] [79] [83]




#stochastics
If the mean of any increment is zero, then the resulting Wiener or Brownian motion process is said to have zero drift.

statusnot read reprioritisations
last reprioritisation on reading queue position [%]
started reading on finished reading on

Stochastic process - Wikipedia
, so it has both continuous index set and states space. [82] But the process can be defined more generally so its state space can be n {\displaystyle n} -dimensional Euclidean space. [71] [79] [83] <span>If the mean of any increment is zero, then the resulting Wiener or Brownian motion process is said to have zero drift. If the mean of the increment for any two points in time is equal to the time difference multiplied by some constant μ {\displaystyle \mu } , w




#stochastics
Almost surely, a sample path of a Wiener process is continuous everywhere but nowhere differentiable. It can be considered a continuous version of the simple random walk.

statusnot read reprioritisations
last reprioritisation on reading queue position [%]
started reading on finished reading on

Stochastic process - Wikipedia
stant μ {\displaystyle \mu } , which is a real number, then the resulting stochastic process is said to have drift μ {\displaystyle \mu } . [84] [85] [86] <span>Almost surely, a sample path of a Wiener process is continuous everywhere but nowhere differentiable. It can be considered a continuous version of the simple random walk. [49] [85] The process arises as the mathematical limit of other stochastic processes such as certain random walks rescaled, [87] [88] which is the subject of Donsker's theorem or inva




#stochastics
If the mean of the increment for any two points in time is equal to the time difference multiplied by some constant , then the resulting stochastic process is said to have drift

statusnot read reprioritisations
last reprioritisation on reading queue position [%]
started reading on finished reading on

Stochastic process - Wikipedia
e space can be n {\displaystyle n} -dimensional Euclidean space. [71] [79] [83] If the mean of any increment is zero, then the resulting Wiener or Brownian motion process is said to have zero drift. <span>If the mean of the increment for any two points in time is equal to the time difference multiplied by some constant μ {\displaystyle \mu } , which is a real number, then the resulting stochastic process is said to have drift μ {\displaystyle \mu } . [84] [85] [86] Almost surely, a sample path of a Wiener process is continuous everywhere but nowhere differentiable. It can be considered a continuous version of the simple rando




#stochastics
If a Poisson process is defined with a single positive constant, then the process is called a homogeneous Poisson process.

statusnot read reprioritisations
last reprioritisation on reading queue position [%]
started reading on finished reading on

Stochastic process - Wikipedia
arameter. This process has the natural numbers as its state space and the non-negative numbers as its index set. This process is also called the Poisson counting process, since it can be interpreted as an example of a counting process. [99] <span>If a Poisson process is defined with a single positive constant, then the process is called a homogeneous Poisson process. [99] [101] The homogeneous Poisson process (in continuous time) is a member of important classes of stochastic processes such as Markov processes and Lévy processes. [49] The homogen




#stochastics
The homogeneous Poisson process can be defined and generalized in different ways. It can be defined such that its index set is the real line, and this stochastic process is also called the stationary Poisson process.

statusnot read reprioritisations
last reprioritisation on reading queue position [%]
started reading on finished reading on

Stochastic process - Wikipedia
constant, then the process is called a homogeneous Poisson process. [99] [101] The homogeneous Poisson process (in continuous time) is a member of important classes of stochastic processes such as Markov processes and Lévy processes. [49] <span>The homogeneous Poisson process can be defined and generalized in different ways. It can be defined such that its index set is the real line, and this stochastic process is also called the stationary Poisson process. [102] [103] If the parameter constant of the Poisson process is replaced with some non-negative integrable function of t {\displaystyle t} ,




#stochastics
If the parameter constant of the Poisson process is replaced with some non-negative integrable function of , the resulting process is called an inhomogeneous or nonhomogeneous Poisson process, where the average density of points of the process is no longer constant.

statusnot read reprioritisations
last reprioritisation on reading queue position [%]
started reading on finished reading on

Stochastic process - Wikipedia
sses. [49] The homogeneous Poisson process can be defined and generalized in different ways. It can be defined such that its index set is the real line, and this stochastic process is also called the stationary Poisson process. [102] [103] <span>If the parameter constant of the Poisson process is replaced with some non-negative integrable function of t {\displaystyle t} , the resulting process is called an inhomogeneous or nonhomogeneous Poisson process, where the average density of points of the process is no longer constant. [104] Serving as a fundamental process in queueing theory, the Poisson process is an important process for mathematical models, where it finds applications for models of events randoml




Flashcard 1735976160524

Tags
#stochastics
Question
Louis Bachelier used [...] to study price changes on the Paris Bourse
Answer


statusnot learnedmeasured difficulty37% [default]last interval [days]               
repetition number in this series0memorised on               scheduled repetition               
scheduled repetition interval               last repetition or drill

Parent (intermediate) annotation

Open it
the Wiener process or Brownian motion process, [a] used by Louis Bachelier to study price changes on the Paris Bourse

Original toplevel document

Stochastic process - Wikipedia
arkets have motivated the extensive use of stochastic processes in finance. [16] [17] [18] Applications and the study of phenomena have in turn inspired the proposal of new stochastic processes. Examples of such stochastic processes include <span>the Wiener process or Brownian motion process, [a] used by Louis Bachelier to study price changes on the Paris Bourse, [21] and the Poisson process, used by A. K. Erlang to study the number of phone calls occurring in a certain period of time. [22] These two stochastic processes are considered the mo







Flashcard 1735977733388

Tags
#stochastics
Question
interpreted as a random element in a function space, a stochastic process can also be called a [...]
Answer
random function


statusnot learnedmeasured difficulty37% [default]last interval [days]               
repetition number in this series0memorised on               scheduled repetition               
scheduled repetition interval               last repetition or drill

Parent (intermediate) annotation

Open it
The term random function is also used to refer to a stochastic or random process, [25] [26] because a stochastic process can also be interpreted as a random element in a function space. <

Original toplevel document

Stochastic process - Wikipedia
are considered the most important and central in the theory of stochastic processes, [1] [4] [23] and were discovered repeatedly and independently, both before and after Bachelier and Erlang, in different settings and countries. [21] [24] <span>The term random function is also used to refer to a stochastic or random process, [25] [26] because a stochastic process can also be interpreted as a random element in a function space. [27] [28] The terms stochastic process and random process are used interchangeably, often with no specific mathematical space for the set that indexes the random variables. [27] [29]







Flashcard 1735979306252

Tags
#stochastics
Question
a stochastic process can be called a random function because it can be interpreted as a [...]
Answer
random element in a function space.

random -- stochastic process
function -- function space


statusnot learnedmeasured difficulty37% [default]last interval [days]               
repetition number in this series0memorised on               scheduled repetition               
scheduled repetition interval               last repetition or drill

Parent (intermediate) annotation

Open it
The term random function is also used to refer to a stochastic or random process, [25] [26] because a stochastic process can also be interpreted as a random element in a function space.

Original toplevel document

Stochastic process - Wikipedia
are considered the most important and central in the theory of stochastic processes, [1] [4] [23] and were discovered repeatedly and independently, both before and after Bachelier and Erlang, in different settings and countries. [21] [24] <span>The term random function is also used to refer to a stochastic or random process, [25] [26] because a stochastic process can also be interpreted as a random element in a function space. [27] [28] The terms stochastic process and random process are used interchangeably, often with no specific mathematical space for the set that indexes the random variables. [27] [29]







Flashcard 1735981141260

Tags
#stochastics
Question
In higher dimensions, stochastic process is usually called [...].
Answer


statusnot learnedmeasured difficulty37% [default]last interval [days]               
repetition number in this series0memorised on               scheduled repetition               
scheduled repetition interval               last repetition or drill

Parent (intermediate) annotation

Open it
If the random variables are indexed by the Cartesian plane or some higher-dimensional Euclidean space, then the collection of random variables is usually called a random field instead.

Original toplevel document

Stochastic process - Wikipedia
hangeably, often with no specific mathematical space for the set that indexes the random variables. [27] [29] But often these two terms are used when the random variables are indexed by the integers or an interval of the real line. [5] [29] <span>If the random variables are indexed by the Cartesian plane or some higher-dimensional Euclidean space, then the collection of random variables is usually called a random field instead. [5] [30] The values of a stochastic process are not always numbers and can be vectors or other mathematical objects. [5] [28] Based on their properties, stochastic processes can be d







Flashcard 1735982714124

Tags
#stochastics
Question
The set used to index the random variables is called the [...].
Answer
index set


statusnot learnedmeasured difficulty37% [default]last interval [days]               
repetition number in this series0memorised on               scheduled repetition               
scheduled repetition interval               last repetition or drill

Parent (intermediate) annotation

Open it
The set used to index the random variables is called the index set.

Original toplevel document

Stochastic process - Wikipedia
stochastic or random process can be defined as a collection of random variables that is indexed by some mathematical set, meaning that each random variable of the stochastic process is uniquely associated with an element in the set. [4] [5] <span>The set used to index the random variables is called the index set. Historically, the index set was some subset of the real line, such as the natural numbers, giving the index set the interpretation of time. [1] Each random variable in the collection t







Flashcard 1735984286988

Tags
#stochastics
Question
Each random variable in the collection takes values from the same mathematical space known as the [...].
Answer
state space


statusnot learnedmeasured difficulty37% [default]last interval [days]               
repetition number in this series0memorised on               scheduled repetition               
scheduled repetition interval               last repetition or drill

Parent (intermediate) annotation

Open it
Each random variable in the collection takes values from the same mathematical space known as the state space.

Original toplevel document

Stochastic process - Wikipedia
element in the set. [4] [5] The set used to index the random variables is called the index set. Historically, the index set was some subset of the real line, such as the natural numbers, giving the index set the interpretation of time. [1] <span>Each random variable in the collection takes values from the same mathematical space known as the state space. This state space can be, for example, the integers, the real line or n {\displaystyle n} -dimensional Euclidean space. [1] [5] An increment i







Flashcard 1735985859852

Tags
#stochastics
Question
Viewed from a function analysis perspective, a single outcome of a stochastic process can be called a [...]
Answer
sample function

Again, remember a function is just a vector with infinite length, and a topology for the notion of proximity and continuity.


statusnot learnedmeasured difficulty37% [default]last interval [days]               
repetition number in this series0memorised on               scheduled repetition               
scheduled repetition interval               last repetition or drill

Parent (intermediate) annotation

Open it
A stochastic process can have many outcomes, due to its randomness, and a single outcome of a stochastic process is called, among other names, a sample function or realization

Original toplevel document

Stochastic process - Wikipedia
r n {\displaystyle n} -dimensional Euclidean space. [1] [5] An increment is the amount that a stochastic process changes between two index values, often interpreted as two points in time. [48] [49] <span>A stochastic process can have many outcomes, due to its randomness, and a single outcome of a stochastic process is called, among other names, a sample function or realization. [28] [50] [imagelink] A single computer-simulated sample function or realization, among other terms, of a three-dimensional Wiener or Brownian motion process for time 0 ≤ t ≤ 2.







Flashcard 1735988219148

Tags
#stochastics
Question
A stochastic process can be classified in different ways, for example, by
  1. its state space,
  2. its index set, or
  3. the [...].
Answer
dependence among the random variables


statusnot learnedmeasured difficulty37% [default]last interval [days]               
repetition number in this series0memorised on               scheduled repetition               
scheduled repetition interval               last repetition or drill

Parent (intermediate) annotation

Open it
A stochastic process can be classified in different ways, for example, by its state space, its index set, or the dependence among the random variables.

Original toplevel document

Stochastic process - Wikipedia
f a three-dimensional Wiener or Brownian motion process for time 0 ≤ t ≤ 2. The index set of this stochastic process is the non-negative numbers, while its state space is three-dimensional Euclidean space. Classifications[edit source] <span>A stochastic process can be classified in different ways, for example, by its state space, its index set, or the dependence among the random variables. One common way of classification is by the cardinality of the index set and the state space. [51] [52] [53] When interpreted as time, if the index set of a stochastic process has a fi







Flashcard 1735989792012

Tags
#stochastics
Question
A stochastic process can be classified in different ways, for example, by
  1. its [...],
  2. its index set, or
  3. the dependence among the random variables.
Answer
state space


statusnot learnedmeasured difficulty37% [default]last interval [days]               
repetition number in this series0memorised on               scheduled repetition               
scheduled repetition interval               last repetition or drill

Parent (intermediate) annotation

Open it
A stochastic process can be classified in different ways, for example, by its state space, its index set, or the dependence among the random variables.

Original toplevel document

Stochastic process - Wikipedia
f a three-dimensional Wiener or Brownian motion process for time 0 ≤ t ≤ 2. The index set of this stochastic process is the non-negative numbers, while its state space is three-dimensional Euclidean space. Classifications[edit source] <span>A stochastic process can be classified in different ways, for example, by its state space, its index set, or the dependence among the random variables. One common way of classification is by the cardinality of the index set and the state space. [51] [52] [53] When interpreted as time, if the index set of a stochastic process has a fi







Flashcard 1735992413452

Tags
#stochastics
Question
the Bernoulli process is just a sequence of [...]
Answer
iid Bernoulli variables.


statusnot learnedmeasured difficulty37% [default]last interval [days]               
repetition number in this series0memorised on               scheduled repetition               
scheduled repetition interval               last repetition or drill

Parent (intermediate) annotation

Open it
One of the simplest stochastic processes is the Bernoulli process, [60] which is a sequence of independent and identically distributed (iid) Bernoulli variables.

Original toplevel document

Stochastic process - Wikipedia
} -dimensional vector process or n {\displaystyle n} -vector process. [51] [52] Examples of stochastic processes[edit source] Bernoulli process[edit source] Main article: Bernoulli process <span>One of the simplest stochastic processes is the Bernoulli process, [60] which is a sequence of independent and identically distributed (iid) random variables, where each random variable takes either the value one or zero, say one with probability p {\displaystyle p} and zero with probability 1 − p {\displaystyle 1-p} . This process can be likened to somebody flipping a coin, where the probability of obtaining a head is p {\displaystyle p} and its value is on







Flashcard 1735993986316

Tags
#stochastics
Question
Random walks are usually defined as [...] of iid random variables or random vectors in Euclidean space
Answer
sums


statusnot learnedmeasured difficulty37% [default]last interval [days]               
repetition number in this series0memorised on               scheduled repetition               
scheduled repetition interval               last repetition or drill

Parent (intermediate) annotation

Open it
Random walks are stochastic processes that are usually defined as sums of iid random variables or random vectors in Euclidean space, so they are processes that change in discrete time.

Original toplevel document

Stochastic process - Wikipedia
one, while the value of a tail is zero. [61] In other words, a Bernoulli process is a sequence of iid Bernoulli random variables, [62] where each coin flip is a Bernoulli trial. [63] Random walk[edit source] Main article: Random walk <span>Random walks are stochastic processes that are usually defined as sums of iid random variables or random vectors in Euclidean space, so they are processes that change in discrete time. [64] [65] [66] [67] [68] But some also use the term to refer to processes that change in continuous time, [69] particularly the Wiener process used in finance, which has led to some c







Flashcard 1735995559180

Tags
#stochastics
Question
Random walks are usually defined as sums of [...] in Euclidean space
Answer
iid random variables or random vectors


statusnot learnedmeasured difficulty37% [default]last interval [days]               
repetition number in this series0memorised on               scheduled repetition               
scheduled repetition interval               last repetition or drill

Parent (intermediate) annotation

Open it
Random walks are stochastic processes that are usually defined as sums of iid random variables or random vectors in Euclidean space, so they are processes that change in discrete time.

Original toplevel document

Stochastic process - Wikipedia
one, while the value of a tail is zero. [61] In other words, a Bernoulli process is a sequence of iid Bernoulli random variables, [62] where each coin flip is a Bernoulli trial. [63] Random walk[edit source] Main article: Random walk <span>Random walks are stochastic processes that are usually defined as sums of iid random variables or random vectors in Euclidean space, so they are processes that change in discrete time. [64] [65] [66] [67] [68] But some also use the term to refer to processes that change in continuous time, [69] particularly the Wiener process used in finance, which has led to some c







Flashcard 1735997132044

Tags
#stochastics
Question
the simple random walk has [...] as the state space
Answer
the integers


statusnot learnedmeasured difficulty37% [default]last interval [days]               
repetition number in this series0memorised on               scheduled repetition               
scheduled repetition interval               last repetition or drill

Parent (intermediate) annotation

Open it
A classic example of a random walk is known as the simple random walk, which is a stochastic process in discrete time with the integers as the state space, and is based on a Bernoulli process, where each iid Bernoulli variable takes either the value positive one or negative one.

Original toplevel document

Stochastic process - Wikipedia
ere are other various types of random walks, defined so their state spaces can be other mathematical objects, such as lattices and groups, and in general they are highly studied and have many applications in different disciplines. [69] [71] <span>A classic example of a random walk is known as the simple random walk, which is a stochastic process in discrete time with the integers as the state space, and is based on a Bernoulli process, where each iid Bernoulli variable takes either the value positive one or negative one. In other words, the simple random walk takes place on the integers, and its value increases by one with probability, say, p {\displaystyle p}







Flashcard 1735998704908

Tags
#stochastics
Question
simple random walk is based on a [...process...]
Answer
Bernoulli process


statusnot learnedmeasured difficulty37% [default]last interval [days]               
repetition number in this series0memorised on               scheduled repetition               
scheduled repetition interval               last repetition or drill

Parent (intermediate) annotation

Open it
A classic example of a random walk is known as the simple random walk, which is a stochastic process in discrete time with the integers as the state space, and is based on a Bernoulli process, where each iid Bernoulli variable takes either the value positive one or negative one.

Original toplevel document

Stochastic process - Wikipedia
ere are other various types of random walks, defined so their state spaces can be other mathematical objects, such as lattices and groups, and in general they are highly studied and have many applications in different disciplines. [69] [71] <span>A classic example of a random walk is known as the simple random walk, which is a stochastic process in discrete time with the integers as the state space, and is based on a Bernoulli process, where each iid Bernoulli variable takes either the value positive one or negative one. In other words, the simple random walk takes place on the integers, and its value increases by one with probability, say, p {\displaystyle p}







Flashcard 1736000802060

Tags
#stochastics
Question
Playing a central role in the theory of probability, [...] is often considered the most important and studied stochastic process,
Answer
the Wiener process


statusnot learnedmeasured difficulty37% [default]last interval [days]               
repetition number in this series0memorised on               scheduled repetition               
scheduled repetition interval               last repetition or drill

Parent (intermediate) annotation

Open it
Playing a central role in the theory of probability, the Wiener process is often considered the most important and studied stochastic process, with connections to other stochastic processes. [1] [2] [3] [78] [79] [80] [81] Its index set and state space are

Original toplevel document

Stochastic process - Wikipedia
wnian motion due to its historical connection as a model for Brownian movement in liquids. [75] [76] [76] [77] [imagelink] Realizations of Wiener processes (or Brownian motion processes) with drift (blue) and without drift (red). <span>Playing a central role in the theory of probability, the Wiener process is often considered the most important and studied stochastic process, with connections to other stochastic processes. [1] [2] [3] [78] [79] [80] [81] Its index set and state space are the non-negative numbers and real numbers, respectively, so it has both continuous index set and states space. [82] But the process can be defined more generally so its state space can be n {\displaystyle n} -dimensional Euclidean space. [71] [79] [83]







Flashcard 1736002374924

Tags
#stochastics
Question
the index set and state space of Wiener process are [...] and [...], respectively
Answer
the non-negative numbers and real numbers


statusnot learnedmeasured difficulty37% [default]last interval [days]               
repetition number in this series0memorised on               scheduled repetition               
scheduled repetition interval               last repetition or drill

Parent (intermediate) annotation

Open it
Playing a central role in the theory of probability, the Wiener process is often considered the most important and studied stochastic process, with connections to other stochastic processes. [1] [2] [3] [78] [79] [80] [81] Its index set and state space are the non-negative numbers and real numbers, respectively, so it has both continuous index set and states space.

Original toplevel document

Stochastic process - Wikipedia
wnian motion due to its historical connection as a model for Brownian movement in liquids. [75] [76] [76] [77] [imagelink] Realizations of Wiener processes (or Brownian motion processes) with drift (blue) and without drift (red). <span>Playing a central role in the theory of probability, the Wiener process is often considered the most important and studied stochastic process, with connections to other stochastic processes. [1] [2] [3] [78] [79] [80] [81] Its index set and state space are the non-negative numbers and real numbers, respectively, so it has both continuous index set and states space. [82] But the process can be defined more generally so its state space can be n {\displaystyle n} -dimensional Euclidean space. [71] [79] [83]







Flashcard 1736004734220

Tags
#stochastics
Question
If [...], then the resulting Wiener or Brownian motion process is said to have zero drift.
Answer
the mean of any increment is zero


statusnot learnedmeasured difficulty37% [default]last interval [days]               
repetition number in this series0memorised on               scheduled repetition               
scheduled repetition interval               last repetition or drill

Parent (intermediate) annotation

Open it
If the mean of any increment is zero, then the resulting Wiener or Brownian motion process is said to have zero drift.

Original toplevel document

Stochastic process - Wikipedia
, so it has both continuous index set and states space. [82] But the process can be defined more generally so its state space can be n {\displaystyle n} -dimensional Euclidean space. [71] [79] [83] <span>If the mean of any increment is zero, then the resulting Wiener or Brownian motion process is said to have zero drift. If the mean of the increment for any two points in time is equal to the time difference multiplied by some constant μ {\displaystyle \mu } , w







Flashcard 1736006307084

Tags
#stochastics
Question
If the mean of any increment is zero, then the resulting Wiener or Brownian motion process is said to have [...].
Answer
zero drift


statusnot learnedmeasured difficulty37% [default]last interval [days]               
repetition number in this series0memorised on               scheduled repetition               
scheduled repetition interval               last repetition or drill

Parent (intermediate) annotation

Open it
If the mean of any increment is zero, then the resulting Wiener or Brownian motion process is said to have zero drift.

Original toplevel document

Stochastic process - Wikipedia
, so it has both continuous index set and states space. [82] But the process can be defined more generally so its state space can be n {\displaystyle n} -dimensional Euclidean space. [71] [79] [83] <span>If the mean of any increment is zero, then the resulting Wiener or Brownian motion process is said to have zero drift. If the mean of the increment for any two points in time is equal to the time difference multiplied by some constant μ {\displaystyle \mu } , w







Flashcard 1736008666380

Tags
#stochastics
Question
If [...] is equal to the time difference multiplied by some constant , then the resulting stochastic process is said to have drift
Answer
the mean of the increment for any two points in time


statusnot learnedmeasured difficulty37% [default]last interval [days]               
repetition number in this series0memorised on               scheduled repetition               
scheduled repetition interval               last repetition or drill

Parent (intermediate) annotation

Open it
If the mean of the increment for any two points in time is equal to the time difference multiplied by some constant , then the resulting stochastic process is said to have drift

Original toplevel document

Stochastic process - Wikipedia
e space can be n {\displaystyle n} -dimensional Euclidean space. [71] [79] [83] If the mean of any increment is zero, then the resulting Wiener or Brownian motion process is said to have zero drift. <span>If the mean of the increment for any two points in time is equal to the time difference multiplied by some constant μ {\displaystyle \mu } , which is a real number, then the resulting stochastic process is said to have drift μ {\displaystyle \mu } . [84] [85] [86] Almost surely, a sample path of a Wiener process is continuous everywhere but nowhere differentiable. It can be considered a continuous version of the simple rando







Flashcard 1736010239244

Tags
#stochastics
Question
If the mean of the increment for any two points in time is equal to [...] , then the resulting stochastic process is said to have drift
Answer
the time difference multiplied by some constant


statusnot learnedmeasured difficulty37% [default]last interval [days]               
repetition number in this series0memorised on               scheduled repetition               
scheduled repetition interval               last repetition or drill

Parent (intermediate) annotation

Open it
If the mean of the increment for any two points in time is equal to the time difference multiplied by some constant , then the resulting stochastic process is said to have drift

Original toplevel document

Stochastic process - Wikipedia
e space can be n {\displaystyle n} -dimensional Euclidean space. [71] [79] [83] If the mean of any increment is zero, then the resulting Wiener or Brownian motion process is said to have zero drift. <span>If the mean of the increment for any two points in time is equal to the time difference multiplied by some constant μ {\displaystyle \mu } , which is a real number, then the resulting stochastic process is said to have drift μ {\displaystyle \mu } . [84] [85] [86] Almost surely, a sample path of a Wiener process is continuous everywhere but nowhere differentiable. It can be considered a continuous version of the simple rando







Flashcard 1736012598540

Tags
#stochastics
Question
If the mean of the increment for any two points in time is equal to the time difference multiplied by some constant , then the resulting stochastic process is said to have [...]
Answer
drift


statusnot learnedmeasured difficulty37% [default]last interval [days]               
repetition number in this series0memorised on               scheduled repetition               
scheduled repetition interval               last repetition or drill

Parent (intermediate) annotation

Open it
If the mean of the increment for any two points in time is equal to the time difference multiplied by some constant , then the resulting stochastic process is said to have drift

Original toplevel document

Stochastic process - Wikipedia
e space can be n {\displaystyle n} -dimensional Euclidean space. [71] [79] [83] If the mean of any increment is zero, then the resulting Wiener or Brownian motion process is said to have zero drift. <span>If the mean of the increment for any two points in time is equal to the time difference multiplied by some constant μ {\displaystyle \mu } , which is a real number, then the resulting stochastic process is said to have drift μ {\displaystyle \mu } . [84] [85] [86] Almost surely, a sample path of a Wiener process is continuous everywhere but nowhere differentiable. It can be considered a continuous version of the simple rando







Flashcard 1736014171404

Tags
#stochastics
Question
Almost surely, a [...] of a Wiener process is continuous everywhere but nowhere differentiable.
Answer
sample path


statusnot learnedmeasured difficulty37% [default]last interval [days]               
repetition number in this series0memorised on               scheduled repetition               
scheduled repetition interval               last repetition or drill

Parent (intermediate) annotation

Open it
Almost surely, a sample path of a Wiener process is continuous everywhere but nowhere differentiable. It can be considered a continuous version of the simple random walk.

Original toplevel document

Stochastic process - Wikipedia
stant μ {\displaystyle \mu } , which is a real number, then the resulting stochastic process is said to have drift μ {\displaystyle \mu } . [84] [85] [86] <span>Almost surely, a sample path of a Wiener process is continuous everywhere but nowhere differentiable. It can be considered a continuous version of the simple random walk. [49] [85] The process arises as the mathematical limit of other stochastic processes such as certain random walks rescaled, [87] [88] which is the subject of Donsker's theorem or inva







Flashcard 1736015744268

Tags
#stochastics
Question
Almost surely, a sample path of a Wiener process is [...property...].
Answer
continuous everywhere but nowhere differentiable


statusnot learnedmeasured difficulty37% [default]last interval [days]               
repetition number in this series0memorised on               scheduled repetition               
scheduled repetition interval               last repetition or drill

Parent (intermediate) annotation

Open it
Almost surely, a sample path of a Wiener process is continuous everywhere but nowhere differentiable. It can be considered a continuous version of the simple random walk.

Original toplevel document

Stochastic process - Wikipedia
stant μ {\displaystyle \mu } , which is a real number, then the resulting stochastic process is said to have drift μ {\displaystyle \mu } . [84] [85] [86] <span>Almost surely, a sample path of a Wiener process is continuous everywhere but nowhere differentiable. It can be considered a continuous version of the simple random walk. [49] [85] The process arises as the mathematical limit of other stochastic processes such as certain random walks rescaled, [87] [88] which is the subject of Donsker's theorem or inva







Flashcard 1736017317132

Tags
#stochastics
Question
Wiener process can be considered a continuous version of [...].
Answer
the simple random walk


statusnot learnedmeasured difficulty37% [default]last interval [days]               
repetition number in this series0memorised on               scheduled repetition               
scheduled repetition interval               last repetition or drill

Parent (intermediate) annotation

Open it
Almost surely, a sample path of a Wiener process is continuous everywhere but nowhere differentiable. It can be considered a continuous version of the simple random walk.

Original toplevel document

Stochastic process - Wikipedia
stant μ {\displaystyle \mu } , which is a real number, then the resulting stochastic process is said to have drift μ {\displaystyle \mu } . [84] [85] [86] <span>Almost surely, a sample path of a Wiener process is continuous everywhere but nowhere differentiable. It can be considered a continuous version of the simple random walk. [49] [85] The process arises as the mathematical limit of other stochastic processes such as certain random walks rescaled, [87] [88] which is the subject of Donsker's theorem or inva







Flashcard 1736018889996

Tags
#stochastics
Question
a homogeneous Poisson process is defined with a [...]
Answer
single positive constant

The constant denotes a fixed area (or length) on the domain.


statusnot learnedmeasured difficulty37% [default]last interval [days]               
repetition number in this series0memorised on               scheduled repetition               
scheduled repetition interval               last repetition or drill

Parent (intermediate) annotation

Open it
If a Poisson process is defined with a single positive constant, then the process is called a homogeneous Poisson process.

Original toplevel document

Stochastic process - Wikipedia
arameter. This process has the natural numbers as its state space and the non-negative numbers as its index set. This process is also called the Poisson counting process, since it can be interpreted as an example of a counting process. [99] <span>If a Poisson process is defined with a single positive constant, then the process is called a homogeneous Poisson process. [99] [101] The homogeneous Poisson process (in continuous time) is a member of important classes of stochastic processes such as Markov processes and Lévy processes. [49] The homogen







Flashcard 1736020462860

Tags
#stochastics
Question
If a Poisson process is defined with a single positive constant, then the process is called a [...].
Answer
homogeneous Poisson process


statusnot learnedmeasured difficulty37% [default]last interval [days]               
repetition number in this series0memorised on               scheduled repetition               
scheduled repetition interval               last repetition or drill

Parent (intermediate) annotation

Open it
If a Poisson process is defined with a single positive constant, then the process is called a homogeneous Poisson process.

Original toplevel document

Stochastic process - Wikipedia
arameter. This process has the natural numbers as its state space and the non-negative numbers as its index set. This process is also called the Poisson counting process, since it can be interpreted as an example of a counting process. [99] <span>If a Poisson process is defined with a single positive constant, then the process is called a homogeneous Poisson process. [99] [101] The homogeneous Poisson process (in continuous time) is a member of important classes of stochastic processes such as Markov processes and Lévy processes. [49] The homogen







Flashcard 1736022035724

Tags
#stochastics
Question
if [...], the homogeneous Poisson process is also called the stationary Poisson process.
Answer
its index set is the real line


statusnot learnedmeasured difficulty37% [default]last interval [days]               
repetition number in this series0memorised on               scheduled repetition               
scheduled repetition interval               last repetition or drill

Parent (intermediate) annotation

Open it
The homogeneous Poisson process can be defined and generalized in different ways. It can be defined such that its index set is the real line, and this stochastic process is also called the stationary Poisson process.

Original toplevel document

Stochastic process - Wikipedia
constant, then the process is called a homogeneous Poisson process. [99] [101] The homogeneous Poisson process (in continuous time) is a member of important classes of stochastic processes such as Markov processes and Lévy processes. [49] <span>The homogeneous Poisson process can be defined and generalized in different ways. It can be defined such that its index set is the real line, and this stochastic process is also called the stationary Poisson process. [102] [103] If the parameter constant of the Poisson process is replaced with some non-negative integrable function of t {\displaystyle t} ,







Flashcard 1736023608588

Tags
#stochastics
Question
The homogeneous Poisson process defined on the real line is called [...].
Answer
the stationary Poisson process


statusnot learnedmeasured difficulty37% [default]last interval [days]               
repetition number in this series0memorised on               scheduled repetition               
scheduled repetition interval               last repetition or drill

Parent (intermediate) annotation

Open it
The homogeneous Poisson process can be defined and generalized in different ways. It can be defined such that its index set is the real line, and this stochastic process is also called the stationary Poisson process.

Original toplevel document

Stochastic process - Wikipedia
constant, then the process is called a homogeneous Poisson process. [99] [101] The homogeneous Poisson process (in continuous time) is a member of important classes of stochastic processes such as Markov processes and Lévy processes. [49] <span>The homogeneous Poisson process can be defined and generalized in different ways. It can be defined such that its index set is the real line, and this stochastic process is also called the stationary Poisson process. [102] [103] If the parameter constant of the Poisson process is replaced with some non-negative integrable function of t {\displaystyle t} ,







Flashcard 1736027540748

Tags
#stochastics
Question
If the parameter constant of the Poisson process is replaced with [...] , the resulting process is called an inhomogeneous or nonhomogeneous Poisson process
Answer
some non-negative integrable function of


statusnot learnedmeasured difficulty37% [default]last interval [days]               
repetition number in this series0memorised on               scheduled repetition               
scheduled repetition interval               last repetition or drill

Parent (intermediate) annotation

Open it
If the parameter constant of the Poisson process is replaced with some non-negative integrable function of , the resulting process is called an inhomogeneous or nonhomogeneous Poisson process, where the average density of points of the process is no longer constant. <

Original toplevel document

Stochastic process - Wikipedia
sses. [49] The homogeneous Poisson process can be defined and generalized in different ways. It can be defined such that its index set is the real line, and this stochastic process is also called the stationary Poisson process. [102] [103] <span>If the parameter constant of the Poisson process is replaced with some non-negative integrable function of t {\displaystyle t} , the resulting process is called an inhomogeneous or nonhomogeneous Poisson process, where the average density of points of the process is no longer constant. [104] Serving as a fundamental process in queueing theory, the Poisson process is an important process for mathematical models, where it finds applications for models of events randoml







Flashcard 1736029900044

Tags
#stochastics
Question
If the constant parameter of the Poisson process is replaced with some non-negative integrable function of , the resulting process is called an [...],
Answer
inhomogeneous or nonhomogeneous Poisson process


statusnot learnedmeasured difficulty37% [default]last interval [days]               
repetition number in this series0memorised on               scheduled repetition               
scheduled repetition interval               last repetition or drill

Parent (intermediate) annotation

Open it
If the parameter constant of the Poisson process is replaced with some non-negative integrable function of , the resulting process is called an inhomogeneous or nonhomogeneous Poisson process, where the average density of points of the process is no longer constant.

Original toplevel document

Stochastic process - Wikipedia
sses. [49] The homogeneous Poisson process can be defined and generalized in different ways. It can be defined such that its index set is the real line, and this stochastic process is also called the stationary Poisson process. [102] [103] <span>If the parameter constant of the Poisson process is replaced with some non-negative integrable function of t {\displaystyle t} , the resulting process is called an inhomogeneous or nonhomogeneous Poisson process, where the average density of points of the process is no longer constant. [104] Serving as a fundamental process in queueing theory, the Poisson process is an important process for mathematical models, where it finds applications for models of events randoml







Flashcard 1736031472908

Tags
#stochastics
Question
With an nonhomogeneous Poisson process, the [...] of points of the process is no longer constant.
Answer
average density

The density is determined by the parameter, obviously.


statusnot learnedmeasured difficulty37% [default]last interval [days]               
repetition number in this series0memorised on               scheduled repetition               
scheduled repetition interval               last repetition or drill

Parent (intermediate) annotation

Open it
body> If the parameter constant of the Poisson process is replaced with some non-negative integrable function of , the resulting process is called an inhomogeneous or nonhomogeneous Poisson process, where the average density of points of the process is no longer constant. <body><html>

Original toplevel document

Stochastic process - Wikipedia
sses. [49] The homogeneous Poisson process can be defined and generalized in different ways. It can be defined such that its index set is the real line, and this stochastic process is also called the stationary Poisson process. [102] [103] <span>If the parameter constant of the Poisson process is replaced with some non-negative integrable function of t {\displaystyle t} , the resulting process is called an inhomogeneous or nonhomogeneous Poisson process, where the average density of points of the process is no longer constant. [104] Serving as a fundamental process in queueing theory, the Poisson process is an important process for mathematical models, where it finds applications for models of events randoml







Flashcard 1741128076556

Tags
#measure-theory #stochastics
Question
Note that P is [...] for each different probability distribution
Answer
a different measure

A measure is a function that maps a set to a non-negative number


statusnot learnedmeasured difficulty37% [default]last interval [days]               
repetition number in this series0memorised on               scheduled repetition               
scheduled repetition interval               last repetition or drill

Parent (intermediate) annotation

Open it
Note that P is a different function for each different probability distri- bution

Original toplevel document (pdf)

cannot see any pdfs







#inner-product-space #vector-space
Functional spaces are generally endowed with additional structure than vector spaces, which may be a topology, allowing the consideration of issues of proximity and continuity.

statusnot read reprioritisations
last reprioritisation on reading queue position [%]
started reading on finished reading on


Parent (intermediate) annotation

Open it
Infinite-dimensional vector spaces arise naturally in mathematical analysis, as function spaces, whose vectors are functions. These vector spaces are generally endowed with additional structure, which may be a topology, allowing the consideration of issues of proximity and continuity. Among these topologies, those that are defined by a norm or inner product are more commonly used, as having a notion of distance between two vectors. This is particularly the

Original toplevel document

Vector space - Wikipedia
roperties, which in some cases can be visualized as arrows. Vector spaces are the subject of linear algebra and are well characterized by their dimension, which, roughly speaking, specifies the number of independent directions in the space. <span>Infinite-dimensional vector spaces arise naturally in mathematical analysis, as function spaces, whose vectors are functions. These vector spaces are generally endowed with additional structure, which may be a topology, allowing the consideration of issues of proximity and continuity. Among these topologies, those that are defined by a norm or inner product are more commonly used, as having a notion of distance between two vectors. This is particularly the case of Banach spaces and Hilbert spaces, which are fundamental in mathematical analysis. Historically, the first ideas leading to vector spaces can be traced back as far as the 17th century's analytic geometry, matrices, systems of linear equations, and Euclidean vectors.




Flashcard 1741386812684

Tags
#inner-product-space #vector-space
Question
A vector space with a topology allows the consideration of issues of [...].
Answer
proximity and continuity


statusnot learnedmeasured difficulty37% [default]last interval [days]               
repetition number in this series0memorised on               scheduled repetition               
scheduled repetition interval               last repetition or drill

Parent (intermediate) annotation

Open it
Functional spaces are generally endowed with additional structure than vector spaces, which may be a topology, allowing the consideration of issues of proximity and continuity.

Original toplevel document

Vector space - Wikipedia
roperties, which in some cases can be visualized as arrows. Vector spaces are the subject of linear algebra and are well characterized by their dimension, which, roughly speaking, specifies the number of independent directions in the space. <span>Infinite-dimensional vector spaces arise naturally in mathematical analysis, as function spaces, whose vectors are functions. These vector spaces are generally endowed with additional structure, which may be a topology, allowing the consideration of issues of proximity and continuity. Among these topologies, those that are defined by a norm or inner product are more commonly used, as having a notion of distance between two vectors. This is particularly the case of Banach spaces and Hilbert spaces, which are fundamental in mathematical analysis. Historically, the first ideas leading to vector spaces can be traced back as far as the 17th century's analytic geometry, matrices, systems of linear equations, and Euclidean vectors.







#inner-product-space #vector-space
Among the topologies of vector spaces, those that are defined by a norm or inner product are more commonly used, as having a notion of distance between two vectors.

statusnot read reprioritisations
last reprioritisation on reading queue position [%]
started reading on finished reading on


Parent (intermediate) annotation

Open it
l analysis, as function spaces, whose vectors are functions. These vector spaces are generally endowed with additional structure, which may be a topology, allowing the consideration of issues of proximity and continuity. <span>Among these topologies, those that are defined by a norm or inner product are more commonly used, as having a notion of distance between two vectors. This is particularly the case of Banach spaces and Hilbert spaces, which are fundamental in mathematical analysis. <span><body><html>

Original toplevel document

Vector space - Wikipedia
roperties, which in some cases can be visualized as arrows. Vector spaces are the subject of linear algebra and are well characterized by their dimension, which, roughly speaking, specifies the number of independent directions in the space. <span>Infinite-dimensional vector spaces arise naturally in mathematical analysis, as function spaces, whose vectors are functions. These vector spaces are generally endowed with additional structure, which may be a topology, allowing the consideration of issues of proximity and continuity. Among these topologies, those that are defined by a norm or inner product are more commonly used, as having a notion of distance between two vectors. This is particularly the case of Banach spaces and Hilbert spaces, which are fundamental in mathematical analysis. Historically, the first ideas leading to vector spaces can be traced back as far as the 17th century's analytic geometry, matrices, systems of linear equations, and Euclidean vectors.




Flashcard 1744145353996

Tags
#inner-product-space #vector-space
Question
Among the topologies of vector spaces, those that are defined by [...] are more commonly used, as having a notion of distance between two vectors.
Answer


statusnot learnedmeasured difficulty37% [default]last interval [days]               
repetition number in this series0memorised on               scheduled repetition               
scheduled repetition interval               last repetition or drill

Parent (intermediate) annotation

Open it
Among the topologies of vector spaces, those that are defined by a norm or inner product are more commonly used, as having a notion of distance between two vectors.

Original toplevel document

Vector space - Wikipedia
roperties, which in some cases can be visualized as arrows. Vector spaces are the subject of linear algebra and are well characterized by their dimension, which, roughly speaking, specifies the number of independent directions in the space. <span>Infinite-dimensional vector spaces arise naturally in mathematical analysis, as function spaces, whose vectors are functions. These vector spaces are generally endowed with additional structure, which may be a topology, allowing the consideration of issues of proximity and continuity. Among these topologies, those that are defined by a norm or inner product are more commonly used, as having a notion of distance between two vectors. This is particularly the case of Banach spaces and Hilbert spaces, which are fundamental in mathematical analysis. Historically, the first ideas leading to vector spaces can be traced back as far as the 17th century's analytic geometry, matrices, systems of linear equations, and Euclidean vectors.







Flashcard 1744147189004

Tags
#inner-product-space #vector-space
Question
Among the topologies of vector spaces, those that are defined by a norm or inner product are more commonly used, as having a notion of [...].
Answer
distance between two vectors
Norm can be understood as the inner product of a vector with itself.


statusnot learnedmeasured difficulty37% [default]last interval [days]               
repetition number in this series0memorised on               scheduled repetition               
scheduled repetition interval               last repetition or drill

Parent (intermediate) annotation

Open it
Among the topologies of vector spaces, those that are defined by a norm or inner product are more commonly used, as having a notion of distance between two vectors.

Original toplevel document

Vector space - Wikipedia
roperties, which in some cases can be visualized as arrows. Vector spaces are the subject of linear algebra and are well characterized by their dimension, which, roughly speaking, specifies the number of independent directions in the space. <span>Infinite-dimensional vector spaces arise naturally in mathematical analysis, as function spaces, whose vectors are functions. These vector spaces are generally endowed with additional structure, which may be a topology, allowing the consideration of issues of proximity and continuity. Among these topologies, those that are defined by a norm or inner product are more commonly used, as having a notion of distance between two vectors. This is particularly the case of Banach spaces and Hilbert spaces, which are fundamental in mathematical analysis. Historically, the first ideas leading to vector spaces can be traced back as far as the 17th century's analytic geometry, matrices, systems of linear equations, and Euclidean vectors.







Flashcard 1744148761868

Tags
#inner-product-space #vector-space
Question
Functional spaces are generally endowed with additional structure than vector spaces, which may be [...], allowing the consideration of issues of proximity and continuity.
Answer


statusnot learnedmeasured difficulty37% [default]last interval [days]               
repetition number in this series0memorised on               scheduled repetition               
scheduled repetition interval               last repetition or drill

Parent (intermediate) annotation

Open it
Functional spaces are generally endowed with additional structure than vector spaces, which may be a topology, allowing the consideration of issues of proximity and continuity.

Original toplevel document

Vector space - Wikipedia
roperties, which in some cases can be visualized as arrows. Vector spaces are the subject of linear algebra and are well characterized by their dimension, which, roughly speaking, specifies the number of independent directions in the space. <span>Infinite-dimensional vector spaces arise naturally in mathematical analysis, as function spaces, whose vectors are functions. These vector spaces are generally endowed with additional structure, which may be a topology, allowing the consideration of issues of proximity and continuity. Among these topologies, those that are defined by a norm or inner product are more commonly used, as having a notion of distance between two vectors. This is particularly the case of Banach spaces and Hilbert spaces, which are fundamental in mathematical analysis. Historically, the first ideas leading to vector spaces can be traced back as far as the 17th century's analytic geometry, matrices, systems of linear equations, and Euclidean vectors.







Flashcard 1756877688076

[unknown IMAGE 1756479753484]
Tags
#has-images #lagrange-multiplier #optimization
Question
At the stationary point there must exist a parameter λ such that [...formula...]
Answer
∇f + λ∇g =0

because ∇f and ∇g are both perpendicular to the equality constraint


statusnot learnedmeasured difficulty37% [default]last interval [days]               
repetition number in this series0memorised on               scheduled repetition               
scheduled repetition interval               last repetition or drill

Parent (intermediate) annotation

Open it
At the stationary point there must exist a parameter λ such that ∇f + λ∇g =0 because ∇f and ∇g are both perpendicular to the equality constraint

Original toplevel document (pdf)

cannot see any pdfs







Flashcard 1756885028108

[unknown IMAGE 1756483423500]
Tags
#Karush-Kuhn-Tucker-condition #has-images
Question
the function f(x) will only be at a maximum if λ satisfies [...formula...]
Answer
\( \lambda > 0 \)


statusnot learnedmeasured difficulty37% [default]last interval [days]               
repetition number in this series0memorised on               scheduled repetition               
scheduled repetition interval               last repetition or drill

Parent (intermediate) annotation

Open it
In optimization with inequality constraint, the sign of the Lagrange multiplier is crucial, because the function f(x) will only be at a maximum if its gradient is oriented away from the region g(x) > 0

Original toplevel document (pdf)

cannot see any pdfs







#inner-product-space #vector-space
A vector space over a field F is a set V together with two operations (the vector addition and scalar multiplication) that satisfy certain axioms.

Elements of V are commonly called vectors. Elements of F are commonly called scalars.

statusnot read reprioritisations
last reprioritisation on reading queue position [%]
started reading on finished reading on

Vector space - Wikipedia
mple above reduces to this one if the arrows are represented by the pair of Cartesian coordinates of their end points. Definition[edit source] In this article, vectors are represented in boldface to distinguish them from scalars. [nb 1] <span>A vector space over a field F is a set V together with two operations that satisfy the eight axioms listed below. The first operation, called vector addition or simply addition + : V × V → V, takes any two vectors v and w and assigns to them a third vector which is commonly written




#inner-product-space #vector-space
A vector space over a field F is a set V together with two operations (the vector addition and scalar multiplication) that satisfy certain axioms.

statusnot read reprioritisations
last reprioritisation on reading queue position [%]
started reading on finished reading on


Parent (intermediate) annotation

Open it
A vector space over a field F is a set V together with two operations (the vector addition and scalar multiplication) that satisfy certain axioms. Elements of V are commonly called vectors. Elements of F are commonly called scalars.

Original toplevel document

Vector space - Wikipedia
mple above reduces to this one if the arrows are represented by the pair of Cartesian coordinates of their end points. Definition[edit source] In this article, vectors are represented in boldface to distinguish them from scalars. [nb 1] <span>A vector space over a field F is a set V together with two operations that satisfy the eight axioms listed below. The first operation, called vector addition or simply addition + : V × V → V, takes any two vectors v and w and assigns to them a third vector which is commonly written




Flashcard 1758240312588

Tags
#inner-product-space #vector-space
Question
[...] is a set V together with two operations (the vector addition and scalar multiplication) that satisfy certain axioms.

Answer
A vector space over a field F


statusnot learnedmeasured difficulty37% [default]last interval [days]               
repetition number in this series0memorised on               scheduled repetition               
scheduled repetition interval               last repetition or drill

Parent (intermediate) annotation

Open it
A vector space over a field F is a set V together with two operations (the vector addition and scalar multiplication) that satisfy certain axioms.

Original toplevel document

Vector space - Wikipedia
mple above reduces to this one if the arrows are represented by the pair of Cartesian coordinates of their end points. Definition[edit source] In this article, vectors are represented in boldface to distinguish them from scalars. [nb 1] <span>A vector space over a field F is a set V together with two operations that satisfy the eight axioms listed below. The first operation, called vector addition or simply addition + : V × V → V, takes any two vectors v and w and assigns to them a third vector which is commonly written







Flashcard 1758241885452

Tags
#inner-product-space #vector-space
Question
In a vector space V over a field F, elements of V are commonly called [...]. Elements of F are commonly called [...] .
Answer
vectors, scalars


statusnot learnedmeasured difficulty37% [default]last interval [days]               
repetition number in this series0memorised on               scheduled repetition               
scheduled repetition interval               last repetition or drill

Parent (intermediate) annotation

Open it
A vector space over a field F is a set V together with two operations (the vector addition and scalar multiplication) that satisfy certain axioms. Elements of V are commonly called vectors. Elements of F are commonly called scalars.

Original toplevel document

Vector space - Wikipedia
mple above reduces to this one if the arrows are represented by the pair of Cartesian coordinates of their end points. Definition[edit source] In this article, vectors are represented in boldface to distinguish them from scalars. [nb 1] <span>A vector space over a field F is a set V together with two operations that satisfy the eight axioms listed below. The first operation, called vector addition or simply addition + : V × V → V, takes any two vectors v and w and assigns to them a third vector which is commonly written







Flashcard 1759676075276

Tags
#spectral-analysis
Question

an eigenfunction of a linear operator D defined on some function space is any non-zero function f in that space that satisfies [...]

Answer
for some scalar eigenvalue λ.


statusnot learnedmeasured difficulty37% [default]last interval [days]               
repetition number in this series0memorised on               scheduled repetition               
scheduled repetition interval               last repetition or drill

Parent (intermediate) annotation

Open it
In mathematics, an eigenfunction of a linear operator D defined on some function space is any non-zero function f in that space that for some scalar eigenvalue λ.

Original toplevel document

Eigenfunction - Wikipedia
ected from Eigenfunction expansion) Jump to: navigation, search [imagelink] This solution of the vibrating drum problem is, at any point in time, an eigenfunction of the Laplace operator on a disk. <span>In mathematics, an eigenfunction of a linear operator D defined on some function space is any non-zero function f in that space that, when acted upon by D, is only multiplied by some scaling factor called an eigenvalue. As an equation, this condition can be written as D f = λ f {\displaystyle Df=\lambda f} for some scalar eigenvalue λ. [1] [2] [3] The solutions to this equation may also be subject to boundary conditions that limit the allowable eigenvalues and eigenfunctions. An eigenfunction is a type of eigenvect







Flashcard 1759925898508

Tags
#topology
Question
A topology must satisfy axioms of [...12...]
Answer
inclusion, complete under infinite union and finit intersection


statusnot learnedmeasured difficulty37% [default]last interval [days]               
repetition number in this series0memorised on               scheduled repetition               
scheduled repetition interval               last repetition or drill

Parent (intermediate) annotation

Open it
A topological space is an ordered pair (X, τ), where X is a set and τ is a collection of subsets of X, satisfying the following axioms: [7] The empty set and X itself belong to τ. Any (finite or infinite) union of members of τ still belongs to τ. The intersection of any finite number of members of τ still belongs to τ.

Original toplevel document

Topological space - Wikipedia
three-point set {1,2,3}. The bottom-left example is not a topology because the union of {2} and {3} [i.e. {2,3}] is missing; the bottom-right example is not a topology because the intersection of {1,2} and {2,3} [i.e. {2}], is missing. <span>A topological space is an ordered pair (X, τ), where X is a set and τ is a collection of subsets of X, satisfying the following axioms: [7] The empty set and X itself belong to τ. Any (finite or infinite) union of members of τ still belongs to τ. The intersection of any finite number of members of τ still belongs to τ. The elements of τ are called open sets and the collection τ is called a topology on X. Examples[edit source] Given X = {1, 2, 3, 4}, the collection τ = {{}, {1, 2, 3, 4}} of only the two subsets of X required by the axioms forms a topology of X, the trivial topology (







Flashcard 1766921473292

Tags
#stochastics
Question
Random walks change in [...] time.
Answer
discrete


statusnot learnedmeasured difficulty37% [default]last interval [days]               
repetition number in this series0memorised on               scheduled repetition               
scheduled repetition interval               last repetition or drill

Parent (intermediate) annotation

Open it
Random walks are stochastic processes that are usually defined as sums of iid random variables or random vectors in Euclidean space, so they are processes that change in discrete time.

Original toplevel document

Stochastic process - Wikipedia
one, while the value of a tail is zero. [61] In other words, a Bernoulli process is a sequence of iid Bernoulli random variables, [62] where each coin flip is a Bernoulli trial. [63] Random walk[edit source] Main article: Random walk <span>Random walks are stochastic processes that are usually defined as sums of iid random variables or random vectors in Euclidean space, so they are processes that change in discrete time. [64] [65] [66] [67] [68] But some also use the term to refer to processes that change in continuous time, [69] particularly the Wiener process used in finance, which has led to some c







Prąd elektryczny jest to zjawisko uporządkowanego ruchu ładunków elektrycznych przez dowolny przekrój poprzeczny środowiska,

statusnot read reprioritisations
last reprioritisation on reading queue position [%]
started reading on finished reading on

pdf

cannot see any pdfs




obwód elektryczny to zbiór elementów połączonych ze sobą w taki sposób, że możliwy jest przepływ prądu elektrycznego co najmniej jedną drogą

statusnot read reprioritisations
last reprioritisation on reading queue position [%]
started reading on finished reading on

pdf

cannot see any pdfs




W skład obwodu elektrycznego wchodzą:
− elementy źródłowe, czyli elementy aktywne wymuszające przepływ prądu,
− elementy odbiorcze, czyli elementy pasywne (rezystory, cewki, kondensatory, silniki, źródła światła itp.), w których energia elektryczna przetwarzana jest w inny rodzaj energii np. w energię cieplną, mechaniczną czy świetlną.

statusnot read reprioritisations
last reprioritisation on reading queue position [%]
started reading on finished reading on

pdf

cannot see any pdfs




Siła elektromotoryczna jest to różnica potencjałów między zaciskami źródła napięcia w warunkach, gdy to źródło nie dostarcza energii elektrycznej do odbiornika.

statusnot read reprioritisations
last reprioritisation on reading queue position [%]
started reading on finished reading on

pdf

cannot see any pdfs




Głównym elementem obwodu jest źródło.

statusnot read reprioritisations
last reprioritisation on reading queue position [%]
started reading on finished reading on

pdf

cannot see any pdfs




Źródło rzeczywiste charakteryzuje się siłą elektromotoryczną E (sem) oraz rezystancją wewnętrzną R w - symbole graficzne źródła napięcia przedstawia rysunek 1.

statusnot read reprioritisations
last reprioritisation on reading queue position [%]
started reading on finished reading on

pdf

cannot see any pdfs




Jeden z zacisków źródła napięcia stałego ma potencjał wyższy – jest to biegun dodatni, oznaczony (+), zaś drugi zacisk ma potencjał niższy i jest to biegun ujemny, oznaczony (-).

statusnot read reprioritisations
last reprioritisation on reading queue position [%]
started reading on finished reading on

pdf

cannot see any pdfs




Biegunowość źródła oznaczamy za pomocą strzałki, której grot wskazuje biegun dodatni.

statusnot read reprioritisations
last reprioritisation on reading queue position [%]
started reading on finished reading on

pdf

cannot see any pdfs




Obwód elektryczny można również określić jako zbiór oczek, przy czym pod pojęciem oczka rozumiemy zbiór połączonych ze sobą gałęzi tworzących zamkniętą drogę dla przepływu prądu, mający te właściwość, że po usunięciu dowolnej gałęzi, pozostałe gałęzie nie tworzą już drogi zamkniętej dla przepływu prądu.

statusnot read reprioritisations
last reprioritisation on reading queue position [%]
started reading on finished reading on

pdf

cannot see any pdfs




#_av #b21 #elektryka #g1 #m_michalski #seo
W elementach obwodu zachodzą trzy rodzaje procesów energetycznych: - wytwarzanie energii (zamiana pewnej energii np. mechanicznej na energię elektryczną) - akumulacja energii - rozpraszanie energi

statusnot read reprioritisations
last reprioritisation on reading queue position [%]
started reading on finished reading on

pdf

cannot see any pdfs




#elektryka
Gałąź obwodu elektrycznego jest utworzona przez jeden lub kilka połączonych ze sobą szeregowo elementów.

statusnot read reprioritisations
last reprioritisation on reading queue position [%]
started reading on finished reading on

pdf

cannot see any pdfs




#elektryka
Jeśli w jednym punkcie zbiegną się co najmniej trzy gałęzie, to w punkcie tym powstanie węzeł obwodu.

statusnot read reprioritisations
last reprioritisation on reading queue position [%]
started reading on finished reading on

pdf

cannot see any pdfs




#elektryka
Jeśli obwód elektryczny zawiera tylko jedną gałąź (jedno oczko), to obwód taki nazywamy obwodem nierozgałęzionym.

statusnot read reprioritisations
last reprioritisation on reading queue position [%]
started reading on finished reading on

pdf

cannot see any pdfs




#elektryka
Jeśli obwód składa się z kilku gałęzi (posiada co najmniej dwa oczka), to obwód taki nazywamy obwodem rozgałęzionym.

statusnot read reprioritisations
last reprioritisation on reading queue position [%]
started reading on finished reading on

pdf

cannot see any pdfs