Mathematical Statistics Quiz#1-2007

Mathematical Statistics Quiz#1-2007 - M ± ሺtሻ, െh...

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Mathematical Statistics Quiz #1 4/12/2007 1. A box contains one two headed coin and eight fair coins. (1) One coin is drawn at random and tossed seven times. Suppose that all seven tosses come up heads. What is the probability that the coin is fair? (2) The coin chosen at random is tossed n times. Given that n heads appear, what is the probability that the coin is fair? What happens to the probability if n՜∞ ? 2. Let X and X have the joint pdf fሺx ,x ሻൌሼ 6x ,0൏x ൏x ൏1 0, otherwise . Find the marginal pdf of X , the conditional pdf of X , given X ൌx , f ଶ୪ଵ ሺx lx , the conditional mean of X , given X ൌx , EሺX lx , EሺX , VሺX . 3. (1) Show that if X is a random variable with the mean µ for which fሺxሻൌ0 for x൏0 , then for any positive constant a , PሺX ൒ aሻ ൑ µ . (2) Let X be a random variable with moment generating function M ± ሺtሻ , െh ൏ ݐ ൏ ݄ . Prove that PሺX൑aሻ൑e ିୟ୲
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Unformatted text preview: M ± ሺtሻ, െh ൏ ݐ ൏ 0 . (3) Prove the statement: if VarሺXሻ ൌ 0 , then PሺX ൌ µሻ ൌ 1 . 4. Let ሺX ଵ , X ଶ ሻ be a random vector such that the variance of X ଶ is finite. Prove that (1) EሾEሺX ଶ |X ଵ ሻሿ ൌ EሺX ଶ ሻ (2) VሾEሺX ଶ |X ଵ ሻሿ ൑ VሺX ଶ ሻ . 5. Let X and Y be independent with moment generating functions M ± ሺtሻ ൌ e ୲ మ and M ± ሺtሻ ൌ e ଶ୲ మ ା୲ . (1) Find the mean and variance of X . (2) Find the moment generating function of X ൅ 2Y ൅ 3 . 6. Let T ൌ ∑ X ୧ N ୧ୀଵ , where ² is a random variable with a finite expectation and the X ୧ are the random variables that are independent of ² and have finite mean EሺXሻ . Show that (1) EሺTሻ ൌ Eሺ²ሻEሺXሻ . (2) VሺTሻ ൌ ሾEሺXሻሿ ଶ Vሺ²ሻ ൅ Eሺ²ሻVሺXሻ ....
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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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