# hw3 - χ 2(1(b Let X 1,X 2,X n iid ∼ N(0 1 ±ind the...

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APPM 4/5520 Problem Set Three (Due Wednesday, September 14th) 1. Let X geom 0 ( p ). (a) Compute the moment generating function of X . Be sure to give and explain restrictions, if any, on the domain of the mgf. (b) Let Y geom 1 ( p ). How does Y relate to X ? Use this relationship to compute the mgf of Y . (Do not compute the mgf for Y “from scratch”.) 2. Let X Γ( α,β ). Use the mgf for X (which you can just take from the table of distributions) to Fnd the mean and variance of X . 3. Let X ij iid unif (0 , 1) for i = 1 , 2 ,... ,I and j = 1 , 2 ,... ,J . (a) ±ind the distribution of Y ij = - 2 ln X ij . (b) ±ind the distribution of Z i = min( Y i 1 ,Y i 2 ,... ,Y i,J ). (c) ±ind the distribution of W = I i =1 Z i . 4. A random variable W is said to have a “chi-squared distribution with n degrees of freedom” if W Γ( n/ 2 , 1 / 2) . We write W χ 2 ( n ) for this speciFc gamma random variable. (a) Use moment generating functions to show that, if X N (0 , 1), then Y = X 2
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Unformatted text preview: χ 2 (1). (b) Let X 1 ,X 2 ,... ,X n iid ∼ N (0 , 1). ±ind the distribution of Y = ∑ n i =1 X 2 i . 5. Let X 1 ,X 2 ,... ,X n be a random sample of size n from a population whose density (pdf) is given by f ( x ) = αx α-1 /θ α · I (0 ,θ ) ( x ) where α > 0 is a known Fxed value, but θ is unknown. Consider the estimator ˆ θ = max( X 1 ,X 2 ,... ,X n ). (a) Show that ˆ θ is a biased estimator of θ . (b) ±ind a multiple of ˆ θ that is an unbiased estimator of θ . (c) ±ind the variance of your estimator from part (b). 6. Required for 5520 Students Only Let X 1 ,X 2 ,... ,X 10 be iid random variables from a distribution with moment generating function M ( t ) = 1 3 b 1-2 3 e t B-1 , for t <-ln(2 / 3) . ±ind P p 10 s i =1 X i > P ....
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