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4703-10-mid-prac

# 4703-10-mid-prac - IEOR E4703 Practice Midterm Exam Fall...

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IEOR E4703: Practice Midterm Exam, Fall 2010. Professor Sigman. 1. X 1 and X 2 are two independent random variables distributed as: P ( X 1 = 0) = 0 . 30 , P ( X 1 = 1) = 0 . 50 , P ( X 1 = 2) = 0 . 20 and P ( X 2 = 1) = 0 . 40 , P ( X 2 = 3) = 0 . 60 (a) Give an algorithm for generating from X = X 1 X 2 that uses two uniform numbers U 1 , U 2 . (b) Give an algorithm for generating from X = X 1 X 2 that uses only one uniform number U . (c) Suppose that X has an exponential distribution at rate λ = 2; F ( x ) = P ( X x ) = 1 - e - 2 x . Suppose you wish to generate a rv Y that has the conditional distribution of X given that it falls in the interval (4 , 10). Show that the following algorithm works: i. Generate V uniformly distributed over the interval ( c, d ) where c = F (4) and d = F (10). ii. Set Y = - (1 / 2) ln(1 - V ). 2. Consider a 2-dimensional BM, X BM ( μ, Σ ), where Σ = σ 2 1 σ 1 σ 2 ρ σ 1 σ 2 ρ σ 2 2 ! . Furthermore, independent of X , consider a compound Poison process, Y ( t ) = N ( t ) X i =1 J i , (1) where { N ( t ) } is a Poisson process at rate λ , and the jumps { J i } are iid distributed as the standard double exponential : f ( x ) = e -|

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