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Flashcard 5574167235852

Tags
#constrained-optimization
Question
Introduce the concept of constrained optimization by using the concepts of contour lines and tangency.
Answer
.


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Flashcard 5584873721100

[unknown IMAGE 5025656081676]
Tags
#axioms-of-probability #has-images #probability-random-variables-statistics-and-random-processes
Question
Describe the axioms of probability and sketch a figure relating the sample space, events and probability.
Answer

Axioms are self-evidently true statements that are unproven, but accepted. Axiomatic probability theory was developed by Andrey N. Kolmogorov, a prominent Russian mathematician, who was the founder of modern probability theory and believed the theory of probability as a mathematical discipline can and should be developed from axioms in exactly the same way as geometry and algebra. In the axiomatic definition of probability, the probability of the event \(A\), denoted by \(P (A)\), in the sample space \(S\) of a random experiment is a real number assigned to \(A\) that satisfies the following axioms of probability:

Axiom I∶ \(P(A) \ge 0\)

Axiom II∶ \(P(S)=1\)

Axiom III∶ If \(A_1 , A_2, \ldots\) is a countable sequence of events such that \(A_i \cap A_j = \emptyset\) for all \(i \neq j\) where \(\emptyset\) is the null event, that is they are pairwise disjoint (mutually exclusive) events, then

\(P \left( A_1 \cup A_2 \cup \ldots \right) = P (A_1) + P (A_2) + \ldots \tag{1.2}\)

Note that these results do not indicate the method of assigning probabilities to the outcomes of an experiment, they merely restrict the way it can be done. These axioms satisfy the intuitive notion of probability. Axiom I of probability highlights that the probability of an event is nonnegative, i.e. chances are always at least zero. AxiomII of probability emphasizes that the probability of all possible outcomes is 1, i.e. the chance that something happens is always 100%. Axiom III of probability underscores that the total probability of a number of disjoint (mutually exclusive) events, i.e. nonoverlapping events, is the sum of the individual probabilities. Figure 1.3 shows the relationship among sample space, events, and probability.


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Axioms are self-evidently true statements that are unproven, but accepted. Axiomatic probability theory was developed by Andrey N. Kolmogorov, a prominent Russian mathematician, who was the founder of modern probability theory and believed the theory of probability as a mathematical discipline can and should be developed from axioms in exactly the same way as geometry and algebra. In the axiomatic definition of probability, the probability of the event \(A\), denoted by \(P (A)\), in the sample space \(S\) of a random experiment is a real number assigned to \(A\) that satisfies the following axioms of probability: Axiom I∶ \(P(A) \ge 0\) Axiom II∶ \(P(S)=1\) Axiom III∶ If \(A_1 , A_2, \ldots\) is a countable sequence of events such that \(A_i \cap A_j = \emptyset\) for all \(i \neq j\) where \(\emptyset\) is the null event, that is they are pairwise disjoint (mutually exclusive) events, then \(P \left( A_1 \cup A_2 \cup \ldots \right) = P (A_1) + P (A_2) + \ldots \tag{1.2}\) Note that these results do not indicate the method of assigning probabilities to the outcomes of an experiment, they merely restrict the way it can be done. These axioms satisfy the intuitive notion of probability. Axiom I of probability highlights that the probability of an event is nonnegative, i.e. chances are always at least zero. AxiomII of probability emphasizes that the probability of all possible outcomes is 1, i.e. the chance that something happens is always 100%. Axiom III of probability underscores that the total probability of a number of disjoint (mutually exclusive) events, i.e. nonoverlapping events, is the sum of the individual probabilities. Figure 1.3 shows the relationship among sample space, events, and probability.

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