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assn7 - Math 361 Problem set 7 Due 1(1.9.6 Let the random...

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Math 361, Problem set 7 Due 10/25/10 1. (1.9.6) Let the random variable X have E [ X ] = μ , E [( X - μ ) 2 ] = σ 2 and mgf M ( t ), - h < t < h . Show that E X - μ σ = 0 , E " X - μ σ 2 # = 1 and E exp t X - μ σ = e - μt/σ M t σ , - hσ < t < hσ. (Recall: exp( x ) = e x ). 2. (1.9.7) Show that the moment generating function of the random variable X having pdf f ( x ) = 1 3 for - 1 < x < 2, zero elsewhere is M ( t ) = e 2 t - e - t 3 t t 6 = 0 1 t = 0 3. (1.9.18) Find the moments of the distribution that has mfg M ( t ) = (1 - t ) - 3 , t < 1. Hint: Find the MacLaurin’s series for M ( t ). 4. (1.9.23) Consider k continuous-type distributions with the following char- acteristics: pdf f i ( x ), mean μ i and variance σ 2 i , i = 1 , 2 , . . . , k . If c i 0, i = 1 , . . . , k and c 1 + · · · + c k = 1, show that the mean and variance of the distribution having pdf
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