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Unformatted text preview: Homework Set No. 2 Probability Theory (235A), Fall 2009 Posted: 10/6/09 Due: 10/13/09 1. (a) Let X be a random variable with distribution function F X and piecewise continuous density function f X . Let [ a,b ] R be an interval (possibly infinite) such that P ( X [ a,b ]) = 1 , and let g : [ a,b ] R be a monotone (strictly) increasing and differentiable function. Prove that the random variable Y = g ( X ) (this is the function on defined by Y ( ) = g ( X ( )), in other words the composition of the two functions g and X ) has density function f Y ( x ) = f X ( g- 1 ( x )) g ( g- 1 ( x )) x ( g ( a ) ,g ( b )) , otherwise. (b) If > 0, we say that a random variable has the exponential distribution with parameter if F X ( x ) = x < , 1- e- x x , and denote this X Exp( ) . Find an algorithm to produce a random variable with Exp( ) distribution using a random number generator that produces uniform random numbers in (0 , 1). In other words, if U U (0 , 1), find a function g : (0 , 1) R such that the random variable X = g ( U ) has distribution Exp( )....
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