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

# Exam 3, 3pm Thurs May 20 2010

Name, RCSID, RIN:




Answering 100 of the 130 points is a complete exam. However you have to mark the questions that you don't want graded with a big X across each one.

1. 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 discrete uniform random variable in the range [0,2]. However, when there is a transmitted signal, the number of photons that the receiver sees is a discrete uniform random variable in the range [1,4]. 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, 3?

2. (5) What is P[signal present|X=k] for k= 0, 1, 2, 3?

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

4. (5) Pretend that T=1. 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?

2. Students applying to RPI:
• Pretend that there are 3,000,000 high school grads this year.
• Each student's decision whether or not to apply to RPI is independent of each other student.
• Each student has a 0.003 probability of applying to RPI.
• The total number of students who apply to RPI is a random variable, X.
1. (10) Name three reasonable probability distributions for X. Write a sentence or two about the advantages and disadvantages of each one, and state which one you prefer for this question and why.

2. (5) What E[X]?

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

4. (5) What is the probability that exactly 9,000 students apply to RPI? It's ok to write a simple expression with numbers, factorial, and e. You don't need to evaluate it.

3. You intend to buy a lottery ticket each week until you win.
• X is the random variable for the number of tickets you will buy.
• The drawings are independent of each other.
• Your probability of winning each drawing is 0.25 .
• Each ticket costs $1. • A winning ticket pays$2 (netting $1 since you had to buy the ticket). • Y is the random variable for your net profit after you finish playing. E.g. if you had to play 4 weeks to get the 1st winning ticket, X=4, Y=-2. 1. (5) What is the name of the appropriate probability distribution for X? 2. (5) What is P[X<=2]? Give a number. 3. (5) What is P[Y>=0]? 4. (5) What E[X]? 5. (5) What E[Y]? 4. You're investigating what Einstein called spooky action at a distance (but you don't need to know what that is to solve this question). For your experiment you and your friend in different cities simultaneously each flip a coin, record the result, and repeat that many times. You record the count for each combo of heads and tails: • you-head, friend-head: 10 (That is, your coin came up heads at the same time as your friend's coin came up heads, 10 times.) • you-head, friend-tail: 20 • you-tail, friend-tail: 60 • you-tail, friend-head: 30 1. (5) What is P[you-head | friend-head]? 2. (5) What is P[you-head]? 3. (5) Are the two coins independent? Justify your answer from the numbers. 5. Horse racing: • Xerxes runs a race in time that is a random variable X that is N(100,10), i.e., normally distributed with mean 100 seconds and standard deviation 10 seconds. • Yggdrasil's time is Y, which is N(110,10). • When Xerxes and Yggdrasil race each other, their times are independent (since this is a course and not the real world). 1. (5) What is P[X>100]? Give a number. 2. (5) What is P[X>110]? Give a number or incomplete integral. 3. (15) What is the probability that Xerxes wins a race with Yggdrasil? Give a number or integral. 6. Darts: • The dart board is a triangle with vertices (0,0), (1,0), (0,1). • The experiment is to throw a dart at the board. • It lands with equal probability anywhere in the board. • The random variables are {$(R,\Theta)$} for the dart's position in polar notation. 1. (5) What is the joint cdf? 2. (5) What is the marginal cdf of {$R$}? 3. (10) Are {$ R$} and {$\Theta\$} independent?

end of exam total: 130 (you pick 100 to be graded).