Probability and statistics

Unit 3 – Homework 3

1. If a random variable X has a uniform density with the parameters, alpha and beta, find its distribution function.

2. Show that if a random variable has a uniform density with the parameters alpha and beta, the rth moment about the mean equals

. 0 when is odd

1 b. when is even

1 2

r

a r

r r

    

  

3. Use the results from Problem 2 to find 3 and 4 for the uniform density with the parameters alpha and beta.

4. If X is a random variable having an exponential distribution with the parameter theta, use Theorems 4.10 and 6.4 to find the moment-generating function of the random variable Y = X –

theta.