final - b. at most 60 were minted in Philadelphia. In each...

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MAT 521 Final June 25, 2004 Instructions: Write the answers and show all your work in the blue books. There are 5 problems. Make sure you do all 5. No books, notes, or collaboration with others. (A Z-table is attached.) Problem 1. (6 points) Five balls are drawn without replacement from an urn containing 3 red balls, 6 black balls, and 1 yellow ball. a. Find the probability that the sample will contain balls of every available color. (Hint: complement rule and inclusion-exclusion.) b. Find the expected number of yellow balls in the sample. Problem 2. (9 points) Let X and Y have joint probability density function given by f ( x, y ) = 2 x y 2 , 0 < x < 1 , y 1 . a. Find the marginal density of Y . b. Find the probability density function of Y . c. Find P ( Y < 2 X ) . Problem 3. (6 points) 68% of all pennies produced in a certain year were coined at the Philadelphia Mint, 28% at the Denver Mint, and the rest at the San Francisco Mint. In a random sample of 100 pennies from the year in question estimate the probability that a. exactly 3 were minted in San Francisco;
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Unformatted text preview: b. at most 60 were minted in Philadelphia. In each case write (a) an expression for the exact (binomial) probability; and (b) the numerical result of either the Poisson or normal approximation, as ap-propriate. 1 Problem 4. (6 points) The moment generating function of X is given by ( t ) = 1 1 + t 2 . a. Find the variance of X. b. Find the moment generating function of 2 X + Y if X and Y are indepen-dent, identically distributed. (Hint: to handle the factor of 2, write down the denition of the moment generating function of X and replace X by 2 X there.) Problem 5 (3 points) Assume that pro golfers scores are normally dis-tributed with mean 70.5 and variance 4.3. After the rst day of play at a golf tournament the eld of 200 players will be downsized to 50 by cutting all those above a certain score. What is the highest cuto score that is likely to achieve this goal? 2...
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This note was uploaded on 02/17/2011 for the course MAT 521 taught by Professor Staff during the Spring '08 term at Syracuse.

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final - b. at most 60 were minted in Philadelphia. In each...

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