hw3 - Homework Set No. 3 Probability Theory (235A), Fall...

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Homework Set No. 3 – Probability Theory (235A), Fall 2011 Due: Tuesday 10/18/11 at discussion section 1. Let X be an exponential r.v. with parameter λ , i.e., F X ( x ) = (1 - e - λx )1 [0 , ) ( x ). Define random variables Y = b X c := sup { n Z : n x } (“the integer part of X ”) , Z = { X } := X - b X c (“the fractional part of X ”) . (a) Compute the (1-dimensional) distributions of Y and Z (in the case of Y , since it’s a discrete random variable it is most convenient to describe the distribution by giving the individual probabilities P ( Y = n ) ,n = 0 , 1 , 2 ,... ; for Z one should compute either the distribution function or density function). (b) Show that Y and Z are independent. (Hint: Check that P ( Y = n,Z t ) = P ( Y = n ) P ( Z t ) for all n and t .) 2. (a) Let X,Y be independent r.v.’s. Define U = min( X,Y ), V = max( X,Y ). Find expressions for the distribution functions F U and F V in terms of the distribution functions of X and Y . (b) Assume that X Exp( λ ) ,Y Exp( μ ) (and are independent as before). Prove that min( X,Y ) has distribution Exp( λ + μ ). Try to give an intuitive explanation in terms of
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hw3 - Homework Set No. 3 Probability Theory (235A), Fall...

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