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

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Unformatted text preview: Homework Set No. 3 Probability Theory (235A), Fall 2009 Posted: 10/13/09 Due: 10/20/09 1. Let X be an exponential r.v. with parameter , i.e., F X ( x ) = (1- e- x )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 its 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 ....
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hw3 - Homework Set No. 3 Probability Theory (235A), Fall...

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