MA519_JAN02 - 1 } and let Z = ( Z 1 ,Z 2 ) be independent...

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QUALIFYING EXAMINATION JANUARY 2002 MATH 519 - Prof. Davis 20 points/problem 1. Let N 1 ,N 2 ,...,N 100 be iid (independent and identically distributed) normal mean 0 and variance 1 variables. Let X 1 ,X 2 ,...,X 100 be iid variables with distribution P ( X i = - 1) = P ( X i =+1)= 1 2 , and let the X i also be independent of the normal variables. Find P ( 100 X i - 1 X i N i > 5). Leave your answer as an integral. 2. The arrival times τ 1 2 < ··· of a Poisson process (with rate λ = 1) are rounded down to the nearest tenth. Let s 1 s 2 ≤ ··· be the numbers this rounding produces. Put N =in f { i : s i is an integer } . (So for example if τ 1 =1 . 86 and τ 2 =3 . 09, then s 1 =1 . 8, s 2 =3, N =2,and s N =3.) (a) Find P ( s N =7). (b) Find N . 3. Let X =( X 1 ,X 2 ) be uniform in the unit square { 0 x 1 , 0
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Unformatted text preview: 1 } and let Z = ( Z 1 ,Z 2 ) be independent of X and have the same distribution. Give the joint density of X + Z . 4. A fair coin is tossed 100 times and then tossed again as many times as tails were obtained in the rst 100 tosses. Let X be the total heads tossed. Find the mean and variance of X in as simplied a form as possible (decimal form is best). 5. (a) Find the density of the median of three independent uniform (0 , 1) variables. (b) Is there a density g such that if X 1 ,X 2 ,X 3 are independent and have density g then the median of the X i has a uniform (0 , 1) distribution?...
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