131A_1_Lecture8-1_Winter_2012

131A_1_Lecture8-1_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 8-1
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UCLA EE131A (KY) 2 Gaussian (Normal) rv (1) Gaussian rv - A Gaussian rv X with two parameters ( , 2 ) has a pdf given by In (1), is called the mean , the standard deviation (sd), and 2 the variance . We will shortly discuss the interpretations of these parameters. The Gaussina rv X occurs so often in theory and applications, it is also called a Normal rv A Normal rv X with the parameters ( , 2 ) is often denoted as X ~ N( , 2 ). 2 2 -(x- μ ) 2 σ 1 f(x) = , - < x < . (1) 2 πσ e 
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UCLA EE131A (KY) 3 Gaussian (Normal) rv (2) In order to show f(x) given in (1) on p. 2 is a valid pdf, we show the following two properties are true: 1. 0 f(x). From (1), we see f(x) can not be negative. 2. Show This turn out to be non-trivial. The “trick” is to show I 2 = 1. Since 0 < I, then I = 1. Let t = (x- )/ and then dt = dx/ . With this change of variable in f(x) of (1), then I now become - I = f(x)dx = 1 . 2 0.5 0.5 -- 2 2 -(x- μ ) -t 2 σ 2 I = (1/2 π )d x / σ =(1/2 π t . ee   
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UCLA EE131A (KY) 4 Gaussian (Normal) rv (3) Then Now, change from rectangular coordinate to polar coordinate to perform the 2-dimensional integral of I 2 . Let t = r sin( ) , s = r cos(( ) , then t 2 +s 2 = r 2 sin 2 ( ) + r 2 cos 2 ( ) = r 2 (sin 2 ( )+ cos 2 ( )) = r 2 , dsdt = dA rect = dA polar = rdrd . 22 ) 20 . 5 0 . 5 -- - - -(t s -t -s 2 II I = ( 1 / 2 π )d t ( 1 / 2 π s = ( 1 / 2 π t d s .
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131A_1_Lecture8-1_Winter_2012 - EE 131A Probability...

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