IEOR E4404 Simulation, Spring 2015
February 6, 2015
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) = t 3 . (You do not need to implement your codes, but you need to present your precise
algorithms and provide clear reasoning.)
Use Monte Carlo simulation to numerically approximate the integral
e −(x+y) sin(xy)dydx.
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
Present a method to generate the random variable X where
P(X = j) =
, j = 1, 2, . . .
Present your algorithm concisely and clearly. You do not need to code your algorithm.
Present a method to generate a random variable X having cumulative distribution func-
x y e −y dy, 0 ≤ x ≤ 1.
F (x) =
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
e −x (n−1)!
, if 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