STAT330 Study Guide - Midterm Guide: Moment-Generating Function, Beta Distribution

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

Document Summary

Problem i: (6pts) if x has a beta distribution with parameters a and b, then the p. d. f. is f (x) = (a) (b) xa 1(1 x)b 1 , 0 < x < 1, a > 0, b > 0 and 0 otherwise. (a) (3pts) find e(x) with a = 3 and b = 3. E(x) = xf (x)dx = (cid:90) 1 (cid:90) 1. (a + 1) (b) (cid:124) a + 1) 1(1 x)b 1dx x(a+1) 1(1 x)b 1 dx (a + 1) (a + b) (a) ((a + 1) + b) (cid:125) density of beta(a + 1, b) = 1 (a + 1) (a + b) (a) (a + 1 + b) a (a) (a + b) (a)(a + b) (a + b) a a + b. Alternative: let a = 3 and b = 3 and evaluate the integral directly. (b) (3pts) find v ar(x) with a = 3 and b = 3 (by rst nding e(x 2)).