Dr. Hackney STA Solutions pg 19

Dr. Hackney STA Solutions pg 19 - 2-4 Solutions Manual for...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 2-4 Solutions Manual for Statistical Inference 2.9 From the probability integral transformation, Theorem 2.1.10, we know that if u(x) = Fx (x), then Fx (X) uniform(0, 1). Therefore, for the given pdf, calculate u(x) = Fx (x) = . 2.10 a. We prove part b), which is equivalent to part a). b. Let Ay = {x : Fx (x) y}. Since Fx is nondecreasing, Ay is a half infinite interval, either open, say (-, xy ), or closed, say (-, xy ]. If Ay is closed, then FY (y) = P (Y y) = P (Fx (X) y) = P (X Ay ) = Fx (xy ) y. The last inequality is true because xy Ay , and Fx (x) y for every x Ay . If Ay is open, then FY (y) = P (Y y) = P (Fx (X) y) = P (X Ay ), as before. But now we have P (X Ay ) = P (X ( - ,xy )) = lim P (X (-, x]), xy 0 if x 1 (x - 1)2 /4 if 1 < x < 3 1 if 3 x Use the Axiom of Continuity, Exercise 1.12, and this equals limxy FX (x) y. The last inequality is true since Fx (x) y for every x Ay , that is, for every x < xy . Thus, FY (y) y for every y. To get strict inequality for some y, let y be a value that is "jumped over" by Fx . That is, let y be such that, for some xy , lim FX (x) < y < FX (xy ). xy For such a y, Ay = (-, xy ), and FY (y) = limxy FX (x) < y. 2.11 a. Using integration by parts with u = x and dv = xe -x2 2 dx then EX 2 = - x2 1 -x2 1 e 2 dx = 2 2 -xe -x2 2 + - - e -x2 2 dx = 1 (2) = 1. 2 Using example 2.1.7 let Y = X 2 . Then -y 1 -y 1 1 1 -y e 2 . fY (y) = e 2 + e 2 = 2 y 2y 2 2 Therefore, EY = 0 -y -y 1 y 1 e 2 dy = -2y 2 e 2 2y 2 + 0 0 y 1 -1 2 e -y 2 1 dy = ( 2) = 1. 2 -y 2 This was obtained using integration by parts with u = 2y 2 and dv = 1 e 2 fY (y) integrates to 1. b. Y = |X| where - < x < . Therefore 0 < y < . Then FY (y) and the fact the = P (Y y) = P (|X| y) = P (-y X y) = P (x y) - P (X -y) = FX (y) - FX (-y). ...
View Full Document

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.

Ask a homework question - tutors are online