Mathematical Statistics Quiz#1SOL-2007

Mathematical Statistics Quiz#1SOL-2007 - Mathematical...

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Unformatted text preview: 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? Sol: Let A : coin tossed is two‐headed, P A A : coin tossed is fair, P A B: 7 heads occurred in 7 tossed, P A B | | | (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 ∞? Sol: lim PA B lim 0 2. Let X and X have the joint pdf6 f x ,x 6x , 0 x x 0, otherwise 1 . x lx , Find the marginal pdf of X , the conditional pdf of X , given X x , f the conditional mean of X , given X x , E X lx , E X , V X . Sol: f x f x lx x 6 x dx 3x , 0 ,0 dx . 6x 1 x ,0 . x 1, x x x , ,0 x 1. 1, E X lx E X f x V X EX · 3x dx 6 x dx EX 3. (1) Show that if X is a random variable with the mean µ for which f x x 0, then for any positive constant a, P X a µ 0 for . Proof: This is Markov inequality (one of home work). (2) Let X be a random variable with moment generating function MX t , h . Prove that P X a e MX t , h 0. Proof: For h 0, P X a P eX e E X e MX t . (3) Prove the statement: if Var X 0, then P X µ 1. Proof: Suppose that P X µ 1. Then, for some ε 0, P |X µ| ε 0. But , by the Chebyshev’s inequality , P |X Therefore, P X µ| µ ε 1. EX µ 0. Which is a contradiction. 4. Let X , X be a random vector such that the variance of X is finite. Prove that EX (1) E E X |X Proof: Done in class. (2) V E X |X V X . Proof: V X EX E E X |X EX E E X |X E X |X E X |X EEX X EEX X V X . e E V X |X Since E V X |X V E X |X 0, we have V E X |X 5. Let X and Y be independent with moment generating functions MX t and MX t e . (1) Find the mean and variance of X. Sol: MX t e · 2t and MX t e · 2t 2e . EX MX 0 0 and V X EX EX (2) Find the moment generating function of X 2Y 3. Sol: MZ t Ee X Y MX 0 2 e MX t MY 2t eee e . 6. Let T ∑N X , where N is a random variable with a finite expectation and the X are the random variables that are independent of N and have finite mean E X . Show that (1) E T E N E X . Proof: E T E ∑N X E E T|N E NE X ENEX E N V X . (2) V T EX VN Proof: Since E T|N NE X and V T|N By 4.(2), we have V T E V T|N E NV X E N V X NV X V E T|N V NE X E X V N . ...
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