An Introduction to Statistical Signal Processing last by Gray Robert

By Gray Robert

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A good way to think of these conditions is that they essentially describe probability measures as set functions defined by either summing or integrating over sets, or by some combination thereof. Hence much of probability theory is simply calculus, especially the evaluation of sums and integrals. To emphasize an important point: a function P which assigns numbers to elements of an event space of a sample space is a probability measure if and only if it satisfies all of the four axioms! A probability space or experiment is a triple (Ω, F, P ) consisting of a sample space Ω, an event space F of subsets of Ω, and a probability measure P defined for all members of F.

A good way to think of these conditions is that they essentially describe probability measures as set functions defined by either summing or integrating over sets, or by some combination thereof. Hence much of probability theory is simply calculus, especially the evaluation of sums and integrals. To emphasize an important point: a function P which assigns numbers to elements of an event space of a sample space is a probability measure if and only if it satisfies all of the four axioms! A probability space or experiment is a triple (Ω, F, P ) consisting of a sample space Ω, an event space F of subsets of Ω, and a probability measure P defined for all members of F.

The probability of the entire sample space is 1: P (Ω) = 1. 8) This follows since integrating 1 over the unit interval yields 1, but it has the intuitive interpretation that the probability that “something happens” is 1. • The probability of the union of disjoint or mutually exclusive regions is the sum of the probabilities of the individual events: If F ∩ G = ∅ , then P (F ∪ G) = P (F ) + P (G). 9) This follows immediately from the properties of integration: P (F ∪ G) = f (r) dr F ∪G = f (r) dr + F f (r) dr G = P (F ) + P (G).

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