# Homework 5, due 2pm Tues Mar 1, 2011

## Notes

1. Email your solutions, as a PDF file, to

wrfranklin+homeworkATgmail.com , replacing AT with @.

1. You may do this homework in teams of two people.
2. Put the homework number, your names and RCS-IDs on the homework. (E.g. I would say something like Homework 3, W. Randolph Franklin, frankwr.)
4. This homework will be graded by Hang.
5. Hang Zhang's office hours are: Mon 7:10-8pm (or as long as people are there asking questions), and Wednesday at 10:00-11:10am in ECSE flip flop lounge.

## Questions

1. The player tosses a fair coin 5 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 5% 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. 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.
4. We are picking two independent discrete random variables, X and Y, uniformly in the interval [0,4]. That means that for both X and Y, S={0,1,2,3,4}.
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]?
5. 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 3 minutes.
1. (5) What is the pmf of X?
2. (5) What is E[X]?
3. (5) What is the probability that X=0?