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

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

## Notes

- Email your solutions, as a PDF file, to

**wrfranklin+homeworkATgmail.com** , replacing **AT** with **@**.

- You may do this homework in teams of two people.
- 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*.) - Please put
**HW05**in your subject line. - This homework will be graded by Hang.
- 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

- The player tosses a fair coin 5 times and counts the number of heads,
k. His reward, Y, is k
^{2}.*(5 pts)*Write the pmf of Y.*(5)*What is E[Y]?*(5)*What is Var[Y]?

- A transmission channel introduces an error 5% of the time. It
transmits 10 messages. Let X be the number of errors.
*(5)*What is the pmf of X?*(5)*What is the probability of 2 or fewer errors?*(5)*What is E[X]?*(5)*What is Var[X]?

- Look at example 3.19 on page 108
*(5)*Redo it, but now with p=1/2.*(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.

- 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}.
*(5)*What is E[X]?*(5)*What is Var[X]?*(5)*What is E[X+Y]?*(5)*What is Var[X+Y]?*(5)*What is E[X|X>Y]?*(5)*What is Var[X|X>Y]?

- 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.
*(5)*What is the pmf of X?*(5)*What is E[X]?*(5)*What is the probability that X=0?