STAT330 Study Guide - Quiz Guide: Ntn1, Marginal Distribution, If And Only If

• School: University of Waterloo
• Department: Statistics
• Course: STAT330
• Professor: Christine Dupont

Document Summary

Assignment 2: since f (x) =(cid:82) x (cid:90) . The region of integration is shown in red below. Switching the order of integra- tion, we have t. Fx(t)dt tfx(t)dt = e(x) (cid:90) (cid:90) t (cid:18)(cid:90) t x. 0 (cid:90) (cid:90) (cid:90) (cid:90) (cid:26) 2. (a) x| = n (0, 1) = fx| (x| ) = 2 exp (cid:27) (cid:26) (cid:90) (cid:90) . 1 exp (cid:26) (cid:18) x2 (cid:122) (cid:125)(cid:124) (cid:123) ( + 1/2) 1 exp d (cid:19) (cid:27) (cid:18) 2 (cid:123)(cid:122) (cid:124) Z(cid:112)x/n (b) let t = where z n (0, 1) and x 2(n): if x 2(n), then mx(t) = e(etx) = (1 2t) n/2. M (t) = e(et ) = e(et x/n) = mx which is the m. g. f. of a gamma(cid:0) n. 2 , 2 n (by the uniqueness theorem). (cid:18) (cid:19) n/2 (cid:19) (cid:18) t (cid:1) density. Alternatively, you can show this by directly nding the p. d. f. : since = x/n, we can write t = z .