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Homework 4

# Homework 4 - X = 1 b log U log q c ∼ Geometric q Here b x...

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Stat 116 Homework 4 Due Wednesday, April 30th. Please show work and justify answers. No credit for a final answer with no explanation, even if the answer is correct. 1. Ross Page 253, problem 27 2. Ross Page 253, problem 18 3. Ross Page 253, problem 21 4. Ross Page 250, problem 36 5. Let X be a random variable such that with probability p (0 , 1) it is distributed N ( μ 1 , σ 1 ) and with probability 1 - p it is distributed N ( μ 2 , σ 2 ). Find P ( a < X < b ) for a = - . 4 , b = 1 . 6 and μ 1 = 0 , μ 2 = 1 , σ 1 = 1 , σ 2 = 4. 6. Let X be a non-negative integer valued r.v. with h ( r ) = P ( X = r | X r ) . If { U i : i 0 } are independent uniform on [0 , 1] show Z = min { n : U n h ( n ) } has the same distribution as X . 7. Let U ∼ U (0 , 1) and 0
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Unformatted text preview: X = 1 + b log U/ log q c ∼ Geometric( q ) . Here b x c stands for the greatest integer less than or equal to x . 8. Let X ∼ Binomial ( n, p ) consider E e-X/n = ( pe-1 /n + 1-p ) n (later on you will see this result via the moment generating function). Let n = 10 and p = 1 / 100 and use the Taylor expansion up to the fourth order to approximate E e-X with functions of the ﬁrst four moments of X , what is your error? ( hint: You may ﬁnd the formulation given in Ch. 4 of Ross for the k th moment of a binomial r.v. via a recursive formula useful. ) 1...
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