131A_1_Lecture16-1_Winter_2012

131A_1_Lecture16-1_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 16-1
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UCLA EE131A (KY) 2 Verifying an empirical pdf (1) We have studied analytical expressions of pdfs of various continuous rv’s and pmf’s of discrete rv’s. From an observed rv data, how do we accept/reject a proposed pdf/pmf for modeling the empirical pdf? In order to use the Chi-Square Test for this purpose, we need to define the Chi-Square rv and its pdf. Let {X 1 , X 2 , …, X n } be n iid Gaussian rv’s of zero mean and unit variance. Then Y = S 2 (n) = X 1 2 + X 2 2 + … + X n 2 , (1) is called a Chi-Square rv with a pdf defined by f Y (y) = (y (n-2)/2 e -y/2 )/(2 n/2 (n/2)), 0 < y < , (2)
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UCLA EE131A (KY) 3 Verifying an empirical pdf (2) where with (m+1) = m! is the factorial for integral m’s. The rv Y in (1) is called a Chi-Square rv with “n-degree of freedom.” This “n” is the number of terms, n, in the sum of (1). We also note, from earlier lectures, we showed that if X ~ N(0, 1), then Y = X 2 has a pdf of f Y (y) = e -y/2 /(2 y) 0.5 , 0 < y . (3) But this Y = X 2 is a special case of a Chi-Square rv of degree one. From (2), we note for n = 1 with (1/2) = 0.5 , then f Y (y) in (2) is equal to f Y (y) in (3). z-1 -x 0 (z) x e dx, 0 < z, 
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UCLA EE131A (KY) 4 Chi-Square Test (1) Consider a rv X with a sample space S X . Consider K disjoint intervals covering all the possible values of
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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_Lecture16-1_Winter_2012 - EE 131A Probability...

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