220finalexam - (f If the gambler plan is to quit playing...

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ISE 220 Final Exam, 12 / 04 all parts are worth 10 points 1. Let X be a Poisson random variable with expected value 2, and let Y be such that P { Y = 1 } = . 3 = 1 - P { Y = 0 } . Suppose that X and Y are independent. Find (a) P { Y > X } (b) P { XY = 0 } (c) E [ XY ] 2. The amount of money a certain gambler wins in a day is a normal random variable with mean 1 and standard deviation 2. Assume that the amounts won on different days are independent. (a) What is the probability the gambler wins a positive amount tomorrow? (b) What is the probability the gambler will win a positive amount of money in exactly three of the next five days? (c) What is the probability that in the next ten days, the gambler will lose money on exactly three of the days, will win between 0 and 2 on exactly five days, and will win more than 2 on exactly two days? (d) Over the next ten days, what is the expected number of days in which the gambler either will lose or will win more than 2? (e) What is the probability the gambler will be ahead after ten days of gambling?
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Unformatted text preview: (f) If the gambler plan is to quit playing after he has had a total of ten winning days, what is the expected number of days he gambles? 3. The density function of X is f ( x ) = c x + x 3 , ≤ x ≤ 1 and is 0 otherwise. (a) Find c . (b) Find the density function of Y = X 2 . (c) Find E [ Y ]. 4. On each transaction, the amount of money withdrawn from an ATM machine is a random variable with mean 10 and variance 4, with the amounts withdrawn in different transactions being independent. (a) What is the expected value of the total amount of money withdrawn in the next 25 transactions? (b) What is the standard deviation of the total amount of money withdrawn in the next 25 transactions? (c) Approximate the probability that the total amount of money withdrawn in the next 25 transactions exceeds 300. 1...
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