STAT333 Lecture Notes - Lecture 4: Gamma Distribution, Binomial Distribution, Exponential Distribution

• School: University of Waterloo
• Department: Statistics
• Course: STAT333
• Professor: Pengfei L I

Document Summary

Stat333 spring 2016 tutorial #4: suppose x1, x2, . are iid random variables from exp( ), the exponential distribution with rate . Let n be geo(p) random variable with 0 < p < 1. We assume that n and all xi"s are independent. Hint: for any given n 1, pn i=1 xi follows a gamma distribution with prob- ability density function ntn 1e t/(n 1)! when t 0 and 0 otherwise. You can directly use this result: (midterm 1 of fall, 2014) let x1, x2, x3 . be independent and identically distributed random variables from binomial distribution with size 2 and probability 0. 5. P (xn = 0) = 0. 25, p (xn = 1) = 0. 5, p (xn = 2) = 0. 25. That is, n is a geometric random variable. X1, x2, x3 . are independent of n. In = (cid:26) 1 if xn = 0. Find e(y ) and v ar(y ): let y be a uniform(0,1) random variable.