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 f_{X|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 i^{th} athlete
is a random variable X_{i}. The random variables X_{1}, X_{2},
... , X_{12} 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.*