ECSE-2500, Engineering Probability, Spring 2010, Rensselaer Polytechnic Institute

# Homework 6, due 2pm Tues Mar 30

Many of these are too hard to do by hand, and require Maple.

1. (10 pts) Let X be a normal random variable with mean=500 and variance=10,000. (This is approximately the distribution of an SAT score.) What is the probability that X>400? X>500? X>600? X>700? X>800?
2. (10 pts) Let X and Y be independent normal random variables each with mean=500 and variance=10,000. What is the probability that X+Y>600? X+Y>1200?
3. Let X be an exponential random variable with parameter {$\lambda=2$} using the definition on page 164 of the book. Maple is inconsistent with the book. .
1. (2) What is its mean?
2. (2) What is its standard deviation?
3. (2) What is its 2nd moment?
4. (2) Verify the formula for variance in terms of the mean and 2nd moment.
5. (2) Plot the pdf of X.
4. (10) Plot the pdfs of the sum of 2, 3, 4, and 5 independent exponential random variables. (They should look more and more like a normal distribution.)
5. The gamma random variable uses the {$\Gamma$} function. Plot {$\Gamma(x)$} for x from 0.5 to 5.