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

Homework 5, due 2pm Tues Mar 23 2010

1. The player tosses a fair coin 4 times and counts the number of heads, k. His reward, Y, is k².
   1. (5 pts) Write the pmf of Y.
   2. (5) What is E[Y]?
   3. (5) What is Var[Y]?
2. A transmission channel introduces an error 10% of the time. It transmits 10 messages. Let X be the number of errors.
   1. (5) What is the pmf of X?
   2. (5) What is the probability of 2 or fewer errors?
   3. (5) What is E[X]?
   4. (5) What is Var[X]?
3. You have an unfair coin, where the probability of heads is 0.4. You toss it 4 times. Let X be the number of heads.
   1. (5) What is the probability of 2 or more heads?
   2. (5) What is E[X]?
   3. (5) What is Var[X]?
4. Look at example 3.19 on page 108
   1. (5) Redo it, but now with p=1/2.
   2. (10) Do a plot of the fraction of packets dropped as function of p, for 0<=p<=1. Use the computer tool of your choice.
5. We are picking two independent random variables, X and Y, uniformly in the interval [0,1].
   1. (5) What is E[X]?
   2. (5) What is Var[X]?
   3. (5) What is E[X+Y]?
   4. (5) What is Var[X+Y]?
   5. (5) What is E[X|X>Y]?
   6. (5) What is Var[X|X>Y]?
6. During the Perseid meteor shower, assume that meteors are independent, and that, on average, there is one per minute. Let X be the number of meteors in 2 minutes.
   1. (5) What is the pmf of X?
   2. (5) What is E[X]?
   3. (5) What is the probability that X=0?
7. (15) Problem 3.88 on page 139.

Total: 145