Econ 310, Spring 2014- Week 5:
Problem 1- EA is marketing its lates TV game. It provides free demo. Suppose a customer who played the demo game buys the full version of the game with probability 0.8. In answering following question, it is safe to assume the behavior of each customer is independent of the others. Use the following probability table for this problem.
|
m
|
P(m)
|
|
0
|
|
|
1
|
0.0064
|
|
2
|
0.0512
|
|
3
|
|
|
4
|
0.4096
|
|
5
|
|
1. Let Xi indicates whether customer i makes a purchase after he played the demo. It takes value 1 if customer i makes the purchase. What are the expected value and variance of Xi?
2. Let M be the number out of the first 5 customers who purchase the full version. Give an expression in terms of the Xi for M. What is the distribution of M?
3. What is the probability that at least 1 consumer buy the full version of the game? What is the probability that exactly 5 consumers buy the full version of the game?
4. What is the probability that the number of consumers, who buy the full version, lies between 1 and 4 (including both 1 and 4)?
5. What are the expected value and variance of the number of customers in these 5 who make a purchase?
6. Sketch the p.d.f and c.d.f.
Problem 2- A bulb factory produces a bulb. Because of some unavoidable reasons, on average, there are 2 defective bulbs for 8 hours of works. Assume that the number of defective bulbs follows Poisson distribution. To calculate the Poisson probability, see the Poisson Probability Table in the last page.
1. Suppose the bulb factory will operate only 4 hours. What is the pdf of the number of defective bulbs in 4 hours of works? Calculate the variance of the number of defective bulbs. What is the probability to have no defective bulb during 4 hours of works?
2. Within 16 hours of works, what is the probability to have exactly 3 defective bulbs?
3. In 24 hours of works, what is the probability to have more than or equal to 10 defective bulbs?
Problem 3- Let X be a random variable with probability density function
f(x) = cx, for 0 < x < 1
1. Is this a discrete or continuous random variable? What value must c take for this to be a proper probability density function?
2. Draw a graph of f(x).
3. Derive the cumulative distribution function for X and sketch a graph?
4. Calculate the following two probabilities: P(0.5) and P(X < 0.5).