05 STAT HWB q discrete rv

# 05 STAT HWB q discrete rv - Dr V.R Bencivenga Economics 329...

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Dr. V.R. Bencivenga Economics 329 PRACTICE HOMEWORK #5B: DISCRETE RANDOM VARIABLES 1. An automobile dealer estimates that of new cars of a certain model sold by the dealership, the distribution of the number of times the car is returned to the dealership with brake problems during the warranty period (X) is as follows: Number of returns 0 1 2 3 4 Proportion of cars sold .28 .36 .23 .09 .04 a. What is the probability that a randomly-chosen new car will be returned more than once during the warranty period? b. What is the probability that a randomly-chosen new car will be returned fewer than three times during the warranty period? c. Suppose you buy a small fleet of ten cars for your business. What is the probability that none of these cars will need to be returned during the warranty period? d. What is the expected number of times a car will need to be returned? The variance of the number of times a car will need to be returned? e. Suppose the cost (in dollars) of servicing the warranty offered on a new car is X 250 100 (the 100 is the fixed cost of the paperwork, and the repair cost per return is 250, in this simplified example). What are the expected cost and the variance of the cost of servicing the warranty on a new car? 2. Invent a 3 x 3 numerical example of a joint probability distribution of two discrete random variables X and Y in which X and Y are uncorrelated but X and Y are statistically dependent . 3. The net present values of two investment projects have the joint probability distribution given in the following table: X = net present value of project #1 5,000 10,000 15,000 3,000 0 .1 .2 Y = net present value 5,000 .1 .2 .1 of project #2 7,000 .2 .1 0 a. What is the expected net present value of project #1? b. What is the variance of the net present value of project #1? c. What is the expected net present value of project #1, given that the net present value of project #2 is 7,000? What is the variance of the net present value of project #1, given that the net present value of project #2 is 7,000? d. What is the covariance between X and Y? (Note : The expression for the covariance has only two non-zero terms.) e. Are X and Y statistically independent? Explain why or why not. f. You invest in both project #1 and project #2. What is the expected net present value of your investment? What is the variance of the net present value of your investment?

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2 4. Find k such that each of the following is a probability function P X (x) of a random variable X: a.
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