Homework 5

# Homework 5 - , 2 , 3 , 4. And the prob-ability of y to take...

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Stat 116 Homework 5 Due Wednesday, May 7th. Please show work and justify answers. No credit for a ﬁnal answer with no explanation, even if the answer is correct. 1. X,Y are independent and exponentially distributed random variables with parameter λ and μ respectively. Let Z = min ( X,Y ), show that Z is independent of I ( X<Y ) . ( I A is an indicator function of set A. I A ( w ) = 1 when w A and zero otherwise) 2. Let I i ,i = 1 , 2 , ··· ,n be independent bernoulli trials with success probability p. Deﬁne X to be the number of success in the ﬁrst 5 trials and Y to be the number of trials needed to see the 6th success. Derive the joint distribution of X,Y. (hint: P ( X = k,Y = l ) = P ( Y = l ) P ( X = k | Y = l ) and derive P ( Y = l ) and P ( X = k | Y = l ) respcetively) 3. ξ and η are independent and both are distributed as standard normal. Find f(x), the probability density function of X = ξ η . 4. y i ,i = 1 , ··· ,n is n independent trials with possible outcome 1
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Unformatted text preview: , 2 , 3 , 4. And the prob-ability of y to take value k is P ( y = k ) = p k . p 1 + p 2 + p 3 + p 4 = 1 ,p k ≥ 0. Let X k be the number of outcomes taking value k . a) Show the joint distribution of X = ( X 1 ,X 2 ,X 3 ,X 4 ). b) Find the joint distribution of ( X 1 ,X 2 ,Y ), where Y = X 3 + X 4 . c) Find the marginal distribution of Y = X 1 + X 2 d)Find the conditional distribution of ( X 1 ,X 2 ) given X 3 ,X 4 . 5. A is a 3x3 matrix 1 2 4 0 2 7 0 0 6 x is a 3x1 vector x = ( x 1 ,x 2 ,x 3 ) T and y is a 3x1 vector (1 , 2 , 3) T . a) Suppose Ax = y , solve for x. b) Derive the matrix A-1 . 6. A is a 2x2 matrix ± 1 ρ ρ 1 ² . Find 2x2 upper trianglar matrix U such that UU T = A . 1...
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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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