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MA519_JAN96

# MA519_JAN96 - 1 and 0< y< 1 g x,y = 0 elsewhere(b Find...

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QUALIFYING EXAMINATION JANUARY 1996 MATH 519 1. Ten balls are thrown randomly into ten boxes. A box can hold any number of balls. Find the mean and variance of the number of empty boxes. 2. A fair six sided die is rolled repeatedly until the first six comes up. Let N i be the number of times the side with i dots comes up, i = 1 , 2 , 3 , 4 , 5. (a) Find the distribution of N 1 . (b) Find the distribution of N 1 + N 2 + N 3 + N 4 + N 5 . (c) Are N 1 and N 1 + N 2 + N 3 + N 4 + N 5 independent. Why? (d) Let Z be the number of those i = 1 , 2 , 3 , 4 , 5 such that N i > 0. Find the distribution of Z . 3. Let X and Y be independent exponential ( λ = 1) variables, that is, they have probability density function f ( t ) = e - t , t > 0. Let Z be independent of ( X, Y ), and suppose P ( Z = 1) = P ( Z = - 1) = 1 2 . (a) Find a function φ ( x, y ) such that φ ( X, Y ) has joint density g ( x, y ) = 1 if
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Unformatted text preview: 1 and 0 < y < 1, g ( x,y ) = 0 elsewhere. (b) Find the joint density of ( ZX,ZY ). (c) Find the density of X X + Y . 4. Let X and Y be independent and identically distributed random variables each with a continuous density f ( t ) which is zero if t 6∈ [0 , 1], and not zero if t ∈ (0 , 1). (a) Find an integer n such that lim ε → 1 ε n P ± | ( X,Y )-± 1 2 , 1 2 ² | ² = δ, where δ ∈ (0 , ∞ ) and | ( a,b )-( c,d ) | is the Euclidean distance between these points. Evaluate δ in terms of f . (b) Find an integer n such that lim ε → 1 ε n P ( | X-Y | < ε ) = δ, where δ ∈ (0 , ∞ ). For which f is this δ minimized?...
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