Re: [問題] Inverse Probability Integral Transfo …
※ 引述《lookinforyou (Loving Kitty =))》之銘言：
: 2. (continuation) If u is uniform(0, 1), deduce that -ln u is exponential(1)
: or gamma(1); and that -2ln u is chi square(2)
用mgf算
x = -ln u
E(e^tx) = E ( e^t(-lnu) ) = E ( e^ln(u^-t) ) = E ( u^-t )
用uniform (0,1) 求出 u^-t的期望值
1 1 u = 1 1
∫ u^-t dx = [ ------ u^(-t-1) ] = ----- = (1-t)^-1
0 -(t-1) u = 0 1-t
由mgf之唯一性知 u~gamma ( 1,1 ) = 指數(1)
第二個
x = -2 ln u = ln u^-2
E(e^tx) = E(e^t(lnu^-2)) = E ( e^ln(u^-2t) ) = E ( u^-2t )
把上面結果來拿用 t代2t mgf = (1-2t)^-1
由mgf之唯一性知 u~gamma ( 1,2 ) = ( 2/2,2 ) ~卡方2
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◆ From: 122.117.204.233
※ 編輯: ksherry 來自: 122.117.204.233 (09/24 04:29)
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09/26 02:26, , 1F
09/26 02:26, 1F
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