Mathematical Statistics Midterm#2-2010

Mathematical Statistics Midterm#2-2010 - Mathematical...

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Mathematical Statistics Midterm #2 5/13/2010 1. Suppose that X~ Bሺn, pሻ and bሺx; n, pሻൌPሺXൌxሻ . (1) Show that bሺx൅1;n ,pሻ ൌ ୮ሺ୬ି୶ሻ ሺ୶ାଵሻሺଵି୮ሻ bሺx; n, pሻ . (2) Show that for pൌ0 .5 the binomial distribution has (a) a maximum at xൌ when n is even; (b) maxima at xൌ ୬ିଵ and xൌ ୬ାଵ when n is odd. 2. Suppose that X and Y are independent Poisson random variables with parameters 1 and 2, respectively. Find (1) PሺX ൌ 1 and Y ൌ 2ሻ ; (2) Pሺ ±ାy ൒1ሻ ; (3) PቀXൌ1ቚ ±ାy ൌ2ቁ . (4) Let X have Poisson λ distribution. Find Eቀ ଵା± . 3. Assume that X has the pdf fሺx; α ሻൌቊ α x ൒ 5 0 otherwise . (1) Find the value of k. What restriction on α is necessary? (2) Find the CDF of X . (3) Find EሺX . What is the restriction on α to ensure the existence of EሺX ? (4) Find the pdf of Yൌlnቀ ±
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