Mathematical Statistics Midterm #2 5/13/2010 1. Suppose that X~ Bሺn, pሻ and bሺx; n, pሻൌPሺXൌxሻ . (1) Show that bሺx1;n ,pሻ ൌ ୮ሺ୬ି୶ሻ ሺ୶ାଵሻሺଵି୮ሻ bሺx; n, pሻ. (2)Show that for pൌ0.5 the binomial distribution has (a) a maximum at xൌ ୬ଶwhen n is even; (b) maxima at xൌ ୬ିଵଶ and xൌ ୬ାଵଶ when n is odd. 2. Suppose that X and Yare independent Poisson random variables with parameters 1 and 2, respectively. Find (1) PሺX ൌ 1 and Y ൌ 2ሻ ; (2) Pሺ ±ାy ଶ 1ሻ ; (3) PቀXൌ1ቚ ±ାy ଶൌ2ቁ. (4) Let X have Poisson ሺ λሻ distribution. Find Eቀ ଵ ଵା±ቁ . 3. Assume that X has the pdf fሺx; α ሻൌቊ ୩ ୶ α x 50 otherwise . (1) Find the value of k. What restriction on α is necessary? (2) Find the CDF of X . (3) Find EሺX ୰ሻ . What is the restriction on α to ensure the existence of EሺX ୰ሻ ? (4) Find the pdf of Yൌlnቀ ± ହ
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