Mathematical Statistics Quiz#2-2007

Mathematical Statistics Quiz#2-2007 - density fሺx ଵ , x...

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Mathematical Statistics Quiz #2 5/31/2007 1. Let X have a geometric distribution with parameter p . (1) Find the distribution function of X and find PሺX ൒ xሻ for the nonnegative integer x . (2) Show that PሺX൐݊൅݉|X൐݉ሻ ൌ PሺX ൐ ݊ሻ . 2. (1) If a random variable X has a Poisson distribution with mean λ . Verify the recursion formula pሺx൅1;λሻ ൌ ୶ାଵ pሺx; λሻ . (2) If a random variable X has a Poisson distribution such that PሺXൌ1ሻൌPሺXൌ3ሻ , find the mode of the distribution. 3. Let X have a gamma distribution with parameters α and β . (1) Find the moment generating function, M ± ሺtሻ , of X . (2) Using M ± ሺtሻ , find the mean and the variance of X . (3) If ሺ1 െ 2tሻ ି଺ ,t൏ , is the moment generating function of the random variable X , find PሺX ൏ 5.23ሻ . 4. Let X and X be two continuous random variables with the joint probability
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Unformatted text preview: density fሺx ଵ , x ଶ ሻ ൌ ൜ 4x ଵ x ଶ , 0 ൏ x ଵ ൏ 1, 0 ൏ x ଶ ൏ 1 0, elsewhere . Let Y ଵ ൌ X ଵ ଶ and Y ଶ ൌ X ଵ X ଶ . (1) Find the joint probability density of Y ଵ and Y ଶ . (2) Let A ൌ ሼሺx ଵ , x ଶ ሻ|0 ൏ x ଵ ൏ 1, 0 ൏ x ଶ ൏ 1ሽ . Draw a picture of the mapping of A. 5. Show that if X is a random variable having the Poisson distribution with the parameter λ and λ ื ∞ , then the moment ‐ generating function of Z ൌ ±ି஛ √஛ approaches the moment ‐ generating function of the standard normal distribution. 6. If Z is a standard normal random variable, find the probability density function of X ൌ Z ଶ ....
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This note was uploaded on 05/09/2011 for the course DIF 1486 taught by Professor Yow-jenjou during the Spring '11 term at National Chiao Tung University.

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