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]?
- (5) What is the probability of 2 or more heads?
- 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?
- (5) What's the pmf of X?
- Let the random variable X be uniform in the interval [-1,1].
Also let Y be a new random variable, defined as Y=X2.
- (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]?
- (5) What's the pdf of X?
- 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?
- (5) What is P[X=k] for k= 0, 1, 2?
- 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]?
- (5) What is E[X]?
end of exam