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

# Exam 2, 2pm ~~Tues April 5~~ Fri April 9 2010

**Name, RCSID, RIN:**

*Answering 100 points is a complete exam. The grader will stop grading if/when you get 100 points.*

- You have an unfair coin, where the probability of heads is 0.3. You toss
it 5 times. Let X be the number of heads.
*(5)*What is the probability of 2 or more heads?

*(5)*What is E[X]?

*(5)*What is Var[X]?

- A call center gets an average of one call per minute.
Let the random variable X be the number of calls in 10 minutes.
*(5)*What's the pmf of X?

*(5)*What's E[X]?

*(5)*What's Var[X]?

*(5)*Write an expression for the cdf of X. (It will be a sum with no closed form.)

*(5)*What's the probability that X=0?

- Let the random variable X be uniform in the interval [-1,1].
Also let Y be a new random variable, defined as Y=X
^{2}.*(5)*What's the pdf of X?

*(5)*What's the cdf of X?

*(5)*What's E[X]?

*(5)*What's Var[X]?

*(5)*What's the pdf of Y?

*(5)*What's the cdf of Y?

*(5)*What's E[Y]?

*(5)*What's Var[Y]?

- A transmitter is sending a signal to a receiver over a noisy channel,
with probability p=1/2. When there is no transmitted signal, the number of
photons that the receiver sees is a Poisson random variable, with
{$\lambda=1$}. However, when there
*is*a transmitted signal, the number of photons that the receiver sees is a Poisson random variable, with {$\lambda=2$}. X is the random variable for the number of photons that the receiver sees.*(5)*What is P[X=k] for k= 0, 1, 2?

*(5)*What is P[signal present|X=k] for k= 0, 1, 2?

*(5)*What is T, the threshhold value of k for which P[signal present|k] >= 1/2?

*(5)*If you use the decision rule that a signal is present if X>=T, then what is the probability that this rule gives the correct answer?

- A non-negative discrete random variable X has probability generating
function {$$G(z)=\frac{z}{2-z}$$}.
*(5)*What is E[X]?

*(5)*What is Var[X]?

*end of exam*