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.
- 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