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final - Problem 4(5 points A mail sorting machine rejects...

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MAT 521 Final July 1, 2005 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 table of standard normal probabilities is attached.) Problem 1. (5 points) Let the moment generating function of a random variable X be given by ψ ( t ) = 1 1 + t . a. Find the variance of X . b. Find the moment generating function of 2 X + 1 . Problem 2. (9 points) Let X and Y have joint probability density function given by f ( x, y ) = e - x y y , 0 < y < 1 , x > 0 . a. Find P ( Y < X ) . b. Identify the marginal distribution of Y . c. Find the probability density function of Y 2 . Problem 3. (6 points) Assume the thicknesses of daily newspapers in a cer- tain city are normally distributed with mean 0 . 25 inches and standard deviation 0 . 1 inches. Assume that thicknesses on different days are independent. Find the probability that tomorrow’s paper will be at least twice as thick as today’s.
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Unformatted text preview: Problem 4. (5 points) A mail sorting machine rejects on average 1% of envelopes as being overweight. Using as model the Poisson distribution, estimate the probability that no more than two envelopes will be rejected in a day’s run of 500 envelopes. 1 Problem 5 (5 points) The Professor likes to keep statistics on the states of the 5 traffic lights he encounters on his drive to work. The results over the last 100 days were: Red: 250; Green: 220; Yellow: 30. Estimate the probability that the Professor will not encounter at least one of the three colors on tomorrow’s drive to work. Hint: Use the inclusion- exclusion formula to find P ( { No Red } ∪ { No Green } ∪ { No Yellow } ) . 2...
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