M 362K, Spring 03, Smith

## Assignment for Friday, December 5

To Hand In:

1. p. 294 #41(a)

2. The joint probability density function of  continuous random variables X and Y is

2    if 0 < x < y < 1
f(x,y) =
0    otherwise.

Find the conditional pdf fX|Y(x|y).

3. A certain elevator can hold 2700 pounds safely. The weight of a random athlete has a normal distribution with mean 225 pounds with standard deviation 25 pounds. What is the probability that  the elevator can safely carry 12 randomly chosen athletes? [Hint: The weight of the ith athlete is a random variable Xi. The random variables X1, X2, ... , X12 are  independent and identically distributed.]

4. One type of lightbulb has lifetime (in months) which is an exponential random variable with parameter 0.1. A second type of lightbulb has lifetime (in months) which is an exponential random variable with parameter 0.5. If you randomly choose a lightbulb of the first type, use it until it burns out, then replace it with a randomly chosen lightbulb of the second type, what is the expected time until both bulbs have burned out?

5. The radius of a cylinder is a random variable uniformly distributed on (0,1). The height of the cylinder is uniformly distributed on (0,2). Find the expected value of the volume of the cylinder. (Added 11/28/03): Assume the height and radius are independent random variables.