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Unformatted text preview: 58 Solutions Manual for Statistical Inference = = = 1 /2 2 (/2)2/2
/2 0 x((+1)/2)1 e(+t 2 +t2 2 )x/2 dx integrand is kernel of gamma((+1)/2, 2/(+t2 ) 1 (( + 1)/2) 2 (/2)2/2 (+1)/2 1 1 ((+1)/2) , 2 (/2) (1 + t /)(+1)/2 the pdf of a t distribution. b. Differentiate both sides with respect to t to obtain fF (t) =
0 yf1 (ty)f (y)dy, where fF is the F pdf. Now write out the two chisquared pdfs and collect terms to get fF (t) = = Now define y = t to get fF (y) =
1/2 ( +1 ) (y/) 2 , (1/2)(/2) (1 + y/)(+1)/2 t1/2 (1/2)(/2)2 t1/2 (1/2)(/2)2
(+1)/2 (+1)/2 0 y (1)/2 e(1+t)y/2 dy ( +1 )2(+1)/2 2 (1 + t)
(+1)/2 . the pdf of an F1, . c. Again differentiate both sides with respect to t, write out the chisquared pdfs, and collect terms to obtain (/m)fF ((/m)t) = tm/2 (m/2)(/2)2
(+m)/2 0 y (m+2)/2 e(1+t)y/2 dy. Now, as before, integrate the gamma kernel, collect terms, and define y = (/m)t to get fF (y) = ( +m ) 2 (m/2)(/2) m m/2 y m/21 (1 + (m/)y)
(+m)/2 , the pdf of an Fm, . 5.21 Let m denote the median. Then, for general n we have P (max(X 1 , . . . , Xn ) > m) = = 5.22 Calculating the cdf of Z 2 , we obtain FZ 2 (z) = = = = = P ((min(X, Y ))2 z) = P (z min(X, Y ) z) P (min(X, Y ) z)  P (min(X, Y )  z) [1  P (min(X, Y ) > z)]  [1  P (min(X, Y ) >  z)] P (min(X, Y ) >  z)  P (min(X, Y ) > z) P (X >  z)P (Y >  z)  P (X > z)P (Y > z), 1  P (Xi m for i = 1, 2, . . . , n) 1  [P (X1 m)]
n n = 1 1 2 . ...
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 Spring '12
 Dr.Hackney
 Statistics

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