Write your name on each assignment. Homework is due at the beginning of class.

THERE WILL BE NO CREDIT FOR LATE HOMEWORK.

- Page 181: Exercises 4.79, 4.89, 4.90
- Page 293: Exercise 6.14
- Page 308: Exercise 6.40
- Let \(Y_1\) and \(Y_2\) be independent random variables with respective probability distributions Poisson(\(\lambda_1\)) and Poisson(\(\lambda_2\)). Let \(Y = Y_1 + Y_2\). Use moment generating functions to show that \(Y\) has probability distribution Poisson(\(\lambda_1 + \lambda_2\)). In other words, if \(Y_1\) and \(Y_2\) are independent Poisson random variables with respective means \(\lambda_1\) and \(\lambda_2\), then the sum \(Y_1 + Y_2\) is a Poisson random variable with mean \(\lambda_1 + \lambda_2\).
- Let \(Y_1\) and \(Y_2\) be independent and identically distributed Exponential random variables with common mean \(E[Y_1]=E[Y_2]=\beta\). Use moment generating functions to show that the sum \(Y = Y_1 + Y_2\) has a Gamma(\(2,\beta\)) probability distribution. What is the probability distribution for the sum of \(n\) independent and identically distributed Exponential(\(\beta\)) random variable \(Y_1, Y_2, \dotsc, Y_n\)?

## Last modified: Thu Jan 17 14:24:11 HST 2019 |