ECSE-2500 Engineering Probability, RPI, Spring 2011 Home Page

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.)
3. Please put HW05 in your subject line.
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?