Lecture8 - Lecture 8 Mariana Olvera-Cravioto Columbia...

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Lecture 8 Mariana Olvera-Cravioto Columbia University [email protected] February 13th, 2012 IEOR 4404, Simulation Lecture 8 1/20
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Generating specific distributions Beta: Let X Beta ( α, β ) , α, β > 0 : f ( x ) = x α - 1 (1 - x ) β - 1 B ( α, β ) , x (0 , 1) Very useful distribution for modeling random probabilities in Bayesian statistics. It can be shown that if W Gamma ( α, θ ) and Y Gamma ( β, θ ) are independent, then X = W W + Y Beta ( α, β ) 1. Generate W Gamma ( α, 1) independent of Y Gamma ( β, 1) . 2. Return X = W W + Y In MATLAB the command is betarnd . IEOR 4404, Simulation Lecture 8 2/20
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The many shapes of the Beta distribution 0 x α = 5, β = 1 α = 1, β = 3 α = β = 0.5 α = 2, β = 5 α = β = 2 1 IEOR 4404, Simulation Lecture 8 3/20
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Generating specific distributions Normal: Note that if X Normal(0,1), then Y = σX + μ Normal ( μ, σ 2 ) , so we only need to be able to generate standard normal random variables. Suppose that X and Y are independent N(0,1) random variables. The joint density of X and Y is f X,Y ( x, y ) = 1 2 π e - x 2 / 2 1 2 π e - y 2 / 2 = 1 2 π e - ( x 2 + y 2 ) / 2 , x, y R Let ( R, Θ) be the polar coordinates of ( X, Y ) , that is, X = R cos Θ and Y = R sin Θ Note that R 2 0 and Θ [0 , 2 π ] . IEOR 4404, Simulation Lecture 8 4/20
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Computing the joint density of R 2 and Θ We can compute the joint density of R 2 and Θ as follows: x y A r r θ P ( R 2 r, Θ θ ) = ZZ A 1 2 π e - ( x 2 + y 2 ) / 2 dx dy = 1 2 π Z r 0 Z θ 0 e - u 2 / 2 u dφ du = θ 2 π Z r 0 e - u 2 / 2 u du = θ 2 π (1 - e - r/ 2 ) where we used the substitution x = u cos φ and y = u sin φ . IEOR 4404, Simulation Lecture 8 5/20
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Generating X and Y Taking derivatives with respect to r and θ we obtain f R 2 , Θ ( r, θ ) = 1 2 π · 1 2 e - r/ 2 , 0 < r < , 0 < θ < 2 π From the joint density we can see that R 2 Exponential(1/2) and Θ Uniform [0 , 2 π ] . Furthermore, R 2 and Θ are independent.
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