assignment_4_08 - ESO 209 PROBABILITY STATISTICS Semester 2...

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ESO 209: PROBABILITY & STATISTICS Semester 2: 2007-08 Assignment #4 Instructor: Amit Mitra [1] Find the expected number of throws of a fair die required to obtain a 6. [2] Consider a sequence of independent coin flips, each of which has a probability p of being heads. Define a random variable as the length of the run (of either heads or tails) started by the first trial. Find X ( ) E X . [3] Find ( ) E X (if it exists) in the following cases: (a) has the p.m.f. X ( ) ( ) ( ) 1 1 if 1,2, .... 0 otherw x x x P X x + = = = ise. (b) has the p.d.f. X ( ) ( ) 2 1 2 if | | 1 0 otherwis x x f x > = e. (c) has the p.d.f. X ( ) 2 1 1 , . 1 f x x x π = − ∞ < + < ∞ [4] Find the mean and variance of the following distributions (a) ( ) 1 , 0 1, 0 a f x a x x a = < < > (b) ( ) 1 , 1,2,..., ; 0 f x x n n n = = > an integer (c) ( ) ( ) 2 3 1 ,0 2 2 f x x x = < < [5] Find the mean and variance of the Weibull random variable having the p.d.f. ( ) 1 exp if 0 otherwise. c c c x x x f x a a a µ µ µ > = Where, and 0, 0 c a > > ( ) , . µ ∈ −∞ ∞ [6] A median of a distribution is a value such that m ( ) 1 2 P X
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