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Annotation 1729336839436

 #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.
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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

Annotation 1732729769228

 #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.
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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

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

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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.

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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

Annotation 1733072915724

 #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.
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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.

Annotation 1733074488588

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

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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

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

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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!

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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.

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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

Annotation 1735875759372

 #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.
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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]

Annotation 1735877856524

 #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.
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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

Annotation 1735879953676

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

Annotation 1735883361548

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

Annotation 1735884934412

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

Annotation 1735886507276

 #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
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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.

Annotation 1735888080140

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

Annotation 1735889653004

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

Annotation 1735959383308

 #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.
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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

Annotation 1735960956172

 #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.
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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}

Annotation 1735963315468

 #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.
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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]

Annotation 1735965412620

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

Annotation 1735966985484

 #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.
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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

Annotation 1735968558348

 #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
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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

Annotation 1735970393356

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

Annotation 1735971966220

 #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.
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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} ,

Annotation 1735973539084

 #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.
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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

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

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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

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

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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. <

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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

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#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

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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.

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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

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

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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.

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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

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

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

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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

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

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

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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

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#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.

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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

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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

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#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

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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.

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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

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#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

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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.

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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

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

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

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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

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

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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.

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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

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

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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.

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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

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

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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.

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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

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

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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.

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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

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#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

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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

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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

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

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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.

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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

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#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

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

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, 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

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#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

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

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, 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

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#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

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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

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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

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#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

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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

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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

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#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

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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

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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

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

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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.

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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

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

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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.

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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

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

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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.

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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

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#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.

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

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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

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

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

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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

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

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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.

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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

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

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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.

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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

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#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

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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. <

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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

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#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

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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.

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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

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#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.

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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>

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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

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#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

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

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Annotation 1741384453388

 #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.
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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

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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

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

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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.

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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.

Annotation 1744140635404

 #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.
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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>

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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

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#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

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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.

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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.

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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.

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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

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#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

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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.

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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

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

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

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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

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Flashcard 1756885028108

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

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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

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Annotation 1758226156812

 #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.
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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

Annotation 1758238739724

 #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.
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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.

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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

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#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

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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

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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.

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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.

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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

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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

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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.

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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

Annotation 1782380104972

 Prąd elektryczny jest to zjawisko uporządkowanego ruchu ładunków elektrycznych przez dowolny przekrój poprzeczny środowiska,
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Annotation 1782383774988

 obwód elektryczny to zbiór elementów połączonych ze sobą w taki sposób, że możliwy jest przepływ prądu elektrycznego co najmniej jedną drogą
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Annotation 1782387707148

 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.
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Annotation 1782391377164

 Siła elektromotoryczna jest to różnica potencjałów między zaciskami źródła napięcia w warunkach, gdy to źródło nie dostarcza energii elektrycznej do odbiornika.
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Annotation 1782392950028

 Głównym elementem obwodu jest źródło.
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Annotation 1782394522892

 Źródło rzeczywiste charakteryzuje się siłą elektromotoryczną E (sem) oraz rezystancją wewnętrzną R w - symbole graficzne źródła napięcia przedstawia rysunek 1.
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Annotation 1782396095756

 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 (-).
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Annotation 1782397668620

 Biegunowość źródła oznaczamy za pomocą strzałki, której grot wskazuje biegun dodatni.
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Annotation 1782399241484

 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.
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Annotation 1782401862924

 #_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
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Annotation 1782407630092

 #elektryka Gałąź obwodu elektrycznego jest utworzona przez jeden lub kilka połączonych ze sobą szeregowo elementów.
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Annotation 1782409202956

 #elektryka Jeśli w jednym punkcie zbiegną się co najmniej trzy gałęzie, to w punkcie tym powstanie węzeł obwodu.
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Annotation 1782410775820

 #elektryka Jeśli obwód elektryczny zawiera tylko jedną gałąź (jedno oczko), to obwód taki nazywamy obwodem nierozgałęzionym.
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Annotation 1782412348684

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