131A_1_Lecture9-2_Winter_2012

131A_1_Lecture9-2_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 9-2
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UCLA EE131A (KY) 2 Second Moment (1) The expectation of a rv X yields the mean . The second moment m 2 is the expectation of X 2 defined by  22 2 - 2 ii i=0 2 - m = E X = x f(x) dx x p , Discrete rv, = x f(x) dx , Continuous rv .
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UCLA EE131A (KY) 3 Second Moment (2) Interpretation of E{X} – Suppose a random voltage is modeled by the rv X. Then the mean = E{X} is the average value of X, which is the DC value of X. Interpretation of E{X 2 } – The second moment is the average power of X. Ex. 1. Second moment of a uniform rv on [0, 1].  22 2 - m = E X = x f(x) dx 1 1 3 2 2 0 0 x m = x f(x) dx = = 1/3 . 3
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UCLA EE131A (KY) 4 Second Moment (3) Ex. 2. Second moment of Bernoulli rv X with S = {0, 1} and P(X=1) = p and P(X=0) = q = 1-p. m 2 = 1 2 p + 0 2 q = p. Ex. 3. Let X be Gaussian with X ~ N( , 2 ). The second moment is given by Take Then 2 2 2 2 -(x- μ ) 2 σ x m = dx 2 πσ e   2 2 2 -t 2 1 μ + σ td t 2 π e  (x- μ )/ σ = t; x = μ + σ t; dx/ σ = dt .
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UCLA EE131A (KY) 5 Ex. 3 (Continue) Use integration by parts, with u = t, du = dt, dv = t exp(-t 2 /2)dt, v = -exp(-t 2
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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_Lecture9-2_Winter_2012 - EE 131A Probability...

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