assignment_4_08

assignment_4_08 - ESO 209: PROBABILITY & STATISTICS...

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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,. ... 0o t h e r w xx x PX x += == i s e . (b) has the p.d.f. X ( ) 2 12 i f ||1 t h e r w i s fx > = e . (c) has the p.d.f. X 2 11 ,. 1 x x π =− < + < [4] Find the mean and variance of the following distributions (a) ( ) 1 ,0 1 , 0 a fx a x x a =< < > (b) 1 , 1,2,. .., ; 0 x nn n > an integer (c) 2 3 1, 0 2 2 x x =− < < [5] Find the mean and variance of the Weibull random variable having the p.d.f. 1 exp if 0 otherwise. cc cx x x aa a µµ µ ⎧⎫ −− ⎪⎪ ⎛⎞ −> ⎨⎬ ⎜⎟ = ⎝⎠ ⎩⎭ Where, and 0, 0 ca >> ( ) ∈−∞∞ [6] A median of a distribution is a value such that m ( ) PX m ≤≥ and ≥≥ . Find the median of the
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assignment_4_08 - ESO 209: PROBABILITY &amp; STATISTICS...

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