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131A_1_Lecture10-1_Winter_2012

131A_1_Lecture10-1_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 10-1
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UCLA EE131A (KY) 2 Expectation of a function g(X) of X (1) So far, we have considered the expectation E{.} of g(X) = X to yields the mean ; g(X) = X 2 to yield the second moment; and g(X) = (X- ) 2 to yield the variance 2 . The n-th moment m n is the expectation of X n defined by We note, using this new notation, the mean = m 1 is the first moment. n n n - m = E X = x f(x) dx .
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UCLA EE131A (KY) 3 Expectation of a function g(X) of X (2) Consider a rv X with a pdf f X (x) and a function g(x). Define a new rv Y = g(X). The expectation of E{Y}= Y can be obtained two ways. Method 1 – Since Y is a rv, it has a pdf f Y (y) and a cdf F Y (y) (considered earlier in Lec8-2). Then Method 2 – Treat Y = g(X) as a function of X. Y Y - μ E{Y} = y f (y) dy . (1) Y X - μ E{g(X)} = g(x) f (x) dx . (2)
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