tutorial9

# tutorial9 - tube; (d) Using the Chebyshev’s inequality,...

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THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT1301 PROBABILITY AND STATISTICS I EXAMPLE CLASS 9 EX.1 Let X Poisson( α ), Y Poisson( β ) be two independent Poisson random variables. Find (a) the moment generating function of X ; (b) the pmf and the moment generating function of X + Y ; (c) the moment generating function of X - Y . 1

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EX.2 The inside diameter of a cylindrical tube is a random variable with a mean of 3 cm and a standard deviation of 0.02 cm, the thickness of the tube is a random variable with a mean of 0.3 cm and a standard deviation 0.005 cm, and the two random variables are independent. (a) Find the moment generating function of X N ( a,b 2 ); (b) Hence, show that if X N ( a,b 2 ) and Y N ( c,d 2 ), where X and Y are inde- pendent, then X + Y N ( a + c,b 2 + d 2 ); (c) Find the mean and the standard deviation of the outside diameter ( W ) of the

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Unformatted text preview: tube; (d) Using the Chebyshev’s inequality, ﬁnd the lower bound for the probability that W is within 0.05 cm from the mean; (e) Evaluate the probability in part (d) by assuming that the inside diameter and the thickness of the tube are independently distributed as normal. 2 EX.3 (a) Given Z ∼ nb ( r,p ) with mgf ± pe t 1-(1-p ) e t ² r , show that if X,Y iid ∼ geom ( p ), then X + Y ∼ nb (2 ,p ). (b) Hence, ﬁnd the conditional pmf of X , given X + Y = n when X and Y are in-dependent and identically distributed geometric random variables with parameter p . 3 EX.4 Suppose the distribution of Y , conditional on X = x , is N ( x,x 2 ) and that the marginal distribution of X is uniform(0,1). Find (a) E( Y ); (b) Var( Y ); and (c) Cov( X,Y ). 4...
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## This note was uploaded on 01/16/2012 for the course STAT 1301 taught by Professor Smslee during the Fall '08 term at HKU.

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tutorial9 - tube; (d) Using the Chebyshev’s inequality,...

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