This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 )....
View
Full
Document
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
 DanRomik
 Probability

Click to edit the document details