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# 05spex2 - Math 431 Second Evening Exam Room 104 Van Hise...

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Unformatted text preview: Math 431 Second Evening Exam Room 104 Van Hise, 4:00pm - 5:00pm, April 8, 2005 M´arton Bal´azs NAME: 1. I have (lots of) 5 dollars bills in my left pocket, and (lots of) 1 dollar bills in my right pocket. At a donation I choose one of my two pockets with equal chance, and then give ten bills from that pocket. (a) (15 points) Compute the expectation and variance of the number of one dollar bills I give at the donation. Answer: Let X be the number of one dollar bills I give. Then X = 10 , with probability 1 2 , , with probability 1 2 . Therefore E ( X ) = 5 and Var ( X ) = 50- 5 2 = 25. (b) (10 points) Compute the expectation and variance of the total amount I give at the dona- tion. Answer: The amount is Y = X + 5(10- X ) = 50- 4 X , hence its expectation is E ( Y ) = 50- 4 · 5 = 30 , and its variance is Var ( Y ) = 4 2 · 25 = 400 . My friend has only one pocket, and he has (lots of) 1 dollar bills and 5 dollars bills in that pocket. He picks ten bills, each being a 1 dollar bill or a 5 dollars bill independently with equal chance. (c) (15 points) Compute the expectation and variance of the number of one dollar bills my friend gives at the donation. Answer: Let X be the number of one dollar bills my friend gives. Then X ∼ Binom(10 , 1 2 ), hence E ( X ) = 10 · 1 2 = 5 and Var ( X ) = 10 · 1 2 · (1- 1 2 ) = 2 . 5. 1 (d) (10 points) Compute the expectation and variance of the total amount my friend gives at the donation....
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05spex2 - Math 431 Second Evening Exam Room 104 Van Hise...

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