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# hw9(1) - (6 Suppose X takes the values 0 1 2 with equal...

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Fall 2011 ORIE 3500/5500 Problem Set 9 Due Friday Nov 4. Reading: Read Chapter 11 Sections 11.1, 11.2; Chapter 14. x/y=page x in course text, problem y. (1) (a) If X, Y are independent and X has density αe - αx , x > 0 and Y has density βe - βx , x > 0 show the density of X + Y is αβ e - αx - e - βx β - α . Note that X + Y must take positive values. (a) Do this directly by computing the convolution of the two densities and then (b) do this a second time using Laplace transforms or moment generating functions. (2) 204/14.6 (3) 204/14.8 Note the “3” in part b was derived in class. (4) 204/14.4 (5) 164/3
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Unformatted text preview: (6) Suppose X takes the values 0 , 1 , 2 with equal values and U ∼ U (0 , 1) and X ⊥⊥ U . What is the distribution function of X + U ? Does X + U have a density? (7) A post oﬃce has two clerks. Service at clerk i is exp( λ i ) distributed. You arrive and require service but both clerks are busy with one customer each, but apart from these customers there is nobody else in front of you waiting for service. Let T be the amount of time you spend in the post oﬃce until your service is completed. Find E ( T ). 1...
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