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131A_1_Lecture8-2_Winter_2012

# 131A_1_Lecture8-2_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 8-2

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UCLA EE131A (KY) 2 Function of a rv (1) A rv is a function defined on the sample space S . Consider a real-valued function g(x) with a real-line domain and a real-line range. Then Y = g(X) is a new rv and is a function that has the domain S and the range of a real-line. To characterize the rv Y, we need to find the cdf or pdf of Y in terms of the cdf or pdf of X and g(x). Method I: F Y (y)=P(s S: Y(s) y)=P(s S: g(X(s)) y) and then f Y (y) = dF Y (y)/dy .
UCLA EE131A (KY) 3 Function of a rv (2) Ex. 1. Consider a uniform rv X on [0, 1] and g(x) = 2x. The new rv Y = g(X) = 2X. We want to find the cdf F Y (y) and pdf f Y (y) of Y. The pdf/cdf of X are known. 0 1 x 1 f(x) X 0 1 x 1 F(x) X  X X 1, 0 x 1, f(x ) = 0, x < 0, 1 < x. 0, x < 0, F (x) = x, 0 x 1, 1, 1 < x.

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UCLA EE131A (KY) 4 Function of a rv (3) Ex. 1 (continue) Use (I): F Y (y) =P(Y y)=P(2X y)= P(X y/2) = F X (y/2) 0, y/2 < 0, y < 0, = y/2, 0
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131A_1_Lecture8-2_Winter_2012 - EE 131A Probability...

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