MA519_JAN05 - X = 1, the 1 coming from the second toss...

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QUALIFYING EXAMINATION JANUARY 2005 MATH 519 - Prof. B. Davis 1 Let X and Y be independent exponential random variables with parameter λ . Let m and M be the smallest and largest of X and Y respectively. Find the joint density of m and M - m . 2 Two fair six sided dice are rolled together until the numbers on both dice are the same. Let X be the number of rolls on which the dice total seven and Y be the number of rolls on which the dice total five. Give the distribution of X , the distribution of X + Y , and the conditional distribution of X given X + Y = 10. Identify these distributions by name if you can. 3 A coin with probability p of heads is tossed one hundred times. Let X be the number of times that a toss was heads and was preceeded by a heads. For example, if the first two tosses are heads and the fifth toss is heads and all other tosses are tails, then
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Unformatted text preview: X = 1, the 1 coming from the second toss being a heads preceeded by a heads. Find the mean and variance of X . Remark: indicator random variables can be helpful. 4 A point is picked at random from the unit disc. Let X be the distance of the point from the center of the disc, Y be the distance of the point from the edge of the disc, and Z be the quadrant the point is in, that is, Z = 1 if both X,Y are bigger than 0, Z = 2 if Y > 0, X < 0, Z = 3 if X,Y < 0, and Z = 4 if X > 0, Y < 0. Find EX , EXY , and EXY Z . 5 Let X k be continuous uniform on (0 ,k ), and let X i , i 1 be independent. Find P ( X n < X n-1 < < X 2 < X 1 )....
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