# Ex4 - X 2 Q4.If two r.v.’s X and Y are independent and...

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THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT 1306: INTRODUCTORY STATISTICS Sem 1 2008/2009 EXAMPLE CLASS 4 Q1. Cast a die a number of independent times until a six appears on the upside of the die. (a) What is the distribution? (b) Find the p.d.f. (c) Show that y =1 f ( y ) = 1 (d) Determine P ( Y = 1 , 3 , 5 , 7 ,...... ) (e) Find the distribution function F ( Y ) = P ( Y y ) 1

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Q2. Let X be a discrete random variable. (a) What is the moment generating function (m.g.f.) of X ? (b) Diﬀerentiate the m.g.f. (with respect to t) and set t=0. Show that this technique gives the moments of X. Q3. If a r.v. X has the m.g.f. M X ( t ) = 1 2 e t + 1 3 e 2 t + 1 6 e 3 t , then ﬁnd the probability function of

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Unformatted text preview: X . 2 Q4.If two r.v.’s X and Y are independent and have probabilities given by P ( X =-1) = 1 4 , P ( X = 0) = 1 2 , P ( X = 2) = 1 4 , and P ( Y = 3) = 1 3 , P ( Y = 4) = 2 3 , (i) Find the m.g.f.s of X and Y . (ii) Find the m.g.f. of Z = X + Y , and use this to derive its probability function. 3 Q5. Suppose that the m.g.f. of X is M X ( t ) = e t / 2 1-e t / 2 . Find the probability function of X . Q6. Let the moments of X be deﬁned by: E ( X r ) = 0 . 8 for r = 1 , 2 , 3 ,...... (a) Find the moment-generating function of X . (b) What is the p.m.f.? (c) What is the distribution? 4...
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## This note was uploaded on 09/06/2010 for the course STAT STAT1306 taught by Professor Prof during the Fall '08 term at HKU.

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Ex4 - X 2 Q4.If two r.v.’s X and Y are independent and...

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