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IEOR E4404 Simulation, Spring 2015
February 6, 2015
Assignment 2
Due date: February 17, 2015
Problem 1. Provide a simulation procedure to generate a random variable X with hazard rate function
given by λ(t) = t3 . (You do not need to implement your codes, but you need to present your precise
algorithms and provide clear reasoning.)
Problem 2.
Use Monte Carlo simulation to numerically approximate the integral
Z
0
∞
Z
x2
e−(x+y) sin(xy)dydx.
0
Attach your codes and the numerical estimates.
Problem 3. A pair of fair dice are to be continually rolled until all possible outcomes 2, 3, . . . , 12 have
occurred at least once. Develop a simulation study to estimate the expected number of dice rolls that
are needed.
Problem 4.
Present a method to generate the random variable X where
j+1 j−1
1
1 2
P(X = j) =
+
, j = 1, 2, . . .
2
2
3j
Present your algorithm concisely and clearly. You do not need to code your algorithm.
Problem 5.
tion
Present a method to generate a random variable X having cumulative distribution funcZ ∞
F (x) =
xy e−y dy, 0 ≤ x ≤ 1.
0
Estimate the mean of X by generating 1000 replications.
Problem 6. A Gamma distribution with parameters (n, 1) (denoted by Gamma(n, 1)) has density
function
(
xn−1
e−x (n−1)!
, if x ≥ 0,
g(x) =
0
if x < 0.
Use the acceptance/rejection method to generate a random variable Y with distribution Gamma(n, 1),
by using an exponential density function
h(x) = λe−λx ,
x ≥ 0.
Find the parameter λ in terms of n that minimizes the expected number of iterations required in the
A/R method.
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