# MATH 472 Homework

## Assignment #1, Due 29 Jan

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

THERE WILL BE NO CREDIT FOR LATE HOMEWORK.

1. Page 181: Exercises 4.79, 4.89, 4.90
2. Page 293: Exercise 6.14
3. Page 308: Exercise 6.40
4. 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 $$\lambda_1 + \lambda_2$$.
5. 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 $$Y_1, Y_2, \dotsc, Y_n$$?