Practice Final

Practice Final - Stat 116 Practice Final 1. Let X1 0 410 X2...

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Stat 116 Practice Final 1. Let X 1 X 2 X 3 Normal 0 0 1 , 4 1 0 1 1 0 0 0 9 (1) Find f X 1 | X 2 ,X 3 ( X 1 | X 2 = 1 , X 3 = 5) (2) Let Y 1 = X 1 + X 2 , Y 2 = X 1 /X 2 . Find the joint p.d.f. of Y 1 , Y 2 : f Y 1 ,Y 2 ( y 1 , y 2 ). 2. Suppose the volume of a raindrop ( V i ) can be modeled as a Gamma distribution with parameter p = 2, λ = 16 (i.e. V i Gamma(2,16), E ( V i ) = 2 / 16). (1) Find the moment generating function (MGF) of the total volume of N raindrops where N is a fixed number (i.e. The MGF of V = Σ N i =1 V i ). (2) Now let N be a random variable with a Poisson distribution with parameter λ = 100. Find E ( V ) and V ar ( V ). (3) What is the moment generating function of V when N Pois(100) (A random sum of random variables)? 3. Let X and Y be independent random variables with X Gamma( r, 1) and Y Gamma( s, 1). Let Z 1 = X + Y and Z 2 = X/ ( X + Y ). (1) Find the joint distribution of
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This note was uploaded on 01/13/2010 for the course STATS 116 taught by Professor Staff during the Spring '07 term at Stanford.

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Practice Final - Stat 116 Practice Final 1. Let X1 0 410 X2...

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