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Unformatted text preview: Homework Set No. 2 – Probability Theory (235A), Fall 2011 Posted: 10/3/11 — Due: 10/11/11 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 Ω defined by Y ( ω ) = g ( X ( ω )), 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 λ > 0, we say that a random variable has the exponential distribution with parameter λ if F X ( x ) = x < , 1- e- λx x ≥ , and denote this X ∼ Exp( λ ) . Find an algorithm to produce a random variable with Exp( λ ) 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( λ )....
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This note was uploaded on 11/13/2011 for the course STAT 235A taught by Professor Danromik during the Fall '11 term at UC Davis.
- Fall '11