131A_1_Lecture9-1_Winter_2012

131A_1_Lecture9-1_Winter_2012 - EE 131A Probability...

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UCLA EE131A (KY) 1 EE 131A Probability Professor Kung Yao Electrical Engineering Department University of California, Los Angeles Lecture 9-1
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UCLA EE131A (KY) 2 Expectation (averaging) (1) The cdf/pdf of a rv X completely characterizes the rv. Often we may want to have a simpler characterization of the rv. The simplest (possibly the most meaningful) characterization of X is its mean or average. Ex. 1. Consider the set of numbers {1, 2, 5, 8}. The average (as we know from “every day” experiences) is = (1 + 2 + 5 + 8) / 4 = 16/4 = 4 . (1) Let X be a discrete rv with the sample space S = {x 1 , …, x n } and a pmf of p i = P(X = x i ), i = 1, …, n. Then the expectation of X (called the mean or the average ) is defined by  n ii i=1 μ = E X = x p .
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UCLA EE131A (KY) 3 Expectation (averaging) (2) Ex. 2. Consider a Bernoulli rv X with S = {0, 1}, P(X = 1) = p, and P(X = 0) = q = 1 – p. Then the mean is = 1 p + 0 q = p . Ex. 3. Consider a discrete uniform rv X with S = {1, 2, 5, 8}. That is, P(X = 1) = P(X = 2) = P(X = 5) = P(X = 8) = ¼ . Then the mean is = 1 (¼) + 2 (¼) + 5 (¼) + 8 (¼) = 16/4 = 4 , which is the same as in (1) for the deterministic average. In fact, the “every day” deterministic average of a sequence of numbers {x 1 , …, x n } is always equal to the average of the discrete uniform rv with the sample space of S = {x 1 , …, x n }, is given by n i i=1 1 μ = x .
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This note was uploaded on 02/29/2012 for the course ELECTRICAL 131a taught by Professor Yao during the Spring '12 term at UCLA.

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131A_1_Lecture9-1_Winter_2012 - EE 131A Probability...

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