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

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

Name, RCSID, RIN:




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

1. 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.
1. (5) What is the probability of 2 or more heads?

2. (5) What is E[X]?

3. (5) What is Var[X]?

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

2. (5) What's E[X]?

3. (5) What's Var[X]?

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

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

3. Let the random variable X be uniform in the interval [-1,1]. Also let Y be a new random variable, defined as Y=X2.
1. (5) What's the pdf of X?

2. (5) What's the cdf of X?

3. (5) What's E[X]?

4. (5) What's Var[X]?

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

6. (5) What's the cdf of Y?

7. (5) What's E[Y]?

8. (5) What's Var[Y]?

4. 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.
1. (5) What is P[X=k] for k= 0, 1, 2?

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

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

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

2. (5) What is Var[X]?

end of exam