STAT330 Study Guide - Quiz Guide: Monotonic Function, If And Only If, Gamma Distribution

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

Document Summary

1. (a) using the identity for an in nite geometric series, 1 (1 p)et assuming (1 p)et < 1 = t < log(1 p) p since the common ratio must be less than 1 for the series to converge. An alternative solution using the property of the p. f. 1 (1 p)et 1 p p. The density in the 2nd line is only properly de ned if the parameter is be- tween 0 and 1. That is, 0 < 1 (1 p)et < 1 = t < log(1 p). 1 (1 p)et , t < log(1 p). (b) E(x 2) = m (0) = (1 p)(2 p) p2 (cid:12)(cid:12)(cid:12)(cid:12)t=0 p (cid:19)2. [1 (1 p)et]2(cid:12)(cid:12)(cid:12)(cid:12)t=0 p p(1 p)et[1 + (1 p)et] V ar(x) = e(x 2) e2(x) = (1 p)(2 p) p2. 2. (a) first note that the support is on (0, ). You can notice this density is a gamma density with = v.