Dr. Hackney STA Solutions pg 35

Dr. Hackney STA Solutions pg 35 - 3-8 Solutions Manual for...

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Unformatted text preview: 3-8 Solutions Manual for Statistical Inference e. The double exponential(, ) pdf is symmetric about . Thus, by Exercise 2.26, EX = . VarX = - (x - )2 1 -|x-|/ e dx = 2 1 z 2 e-|z| dz 2 - = 2 0 z 2 e-z dz = 2 (3) = 2 2 . 3.23 a. x--1 dx = -1 - x = 1 , thus f (x) integrates to 1 . b. EX n = n (n-) , therefore EX EX 2 VarX = = = (1 - ) 2 (2 - ) 2 () - 2 2- (1-) 2 c. If < 2 the integral of the second moment is infinite. 1 3.24 a. fx (x) = e-x/ , x > 0. For Y = X 1/ , fY (y) = tion z = y /, we calculate -y / -1 y , e y > 0. Using the transforma- EY n = y +n-1 e-y 0 / dy = n/ 0 2 +1 z n/ e-z dz = n/ -2 2 n +1 . 1 Thus EY = 1/ ( + 1) and VarY = 2/ 1 +1 /2 . b. fx (x) = 1 -x/ , e x > 0. For Y = (2X/)1/2 , fY (y) = ye-y , y > 0 . We now notice that y e 2 -y 2 /2 EY = 0 2 dy = 2 2 since 1 - y 2 e-y /2 = 1, the variance of a standard normal, and the integrand is sym2 metric. Use integration-by-parts to calculate the second moment EY 2 = 0 y 3 e-y 2 2 /2 dy = 2 0 ye-y 2 /2 dy = 2, where we take u = y 2 , dv = ye-y c. The gamma(a, b) density is /2 . Thus VarY = 2(1 - /4). 1 xa-1 e-x/b . (a)ba fX (x) = Make the transformation y = 1/x with dx = -dy/y 2 to get fY (y) = fX (1/y)|1/y 2 | = 1 (a)ba 1 y a+1 e-1/by . ...
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This note was uploaded on 02/03/2012 for the course STA 1014 taught by Professor Dr.hackney during the Spring '12 term at UNF.

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