ISYE 6644 ASY OAN SIMULATION FINAL
COMPREHENSIVE SCRIPT WITH SOLUTIONS
◉ Suppose that Z1, Z2, and Z3 are i.i.d. Nor(0,1) random variables,
and let T = Z1 /sqrt((Z 2 2 + Z 2 3 )/2) . Find the value of x such that
Pr(T < x) = 0.99.
Answer: x=6.965
◉ Suppose X has the Laplace distribution with p.d.f. f(x) = λ/2
e^−λ|x| for x ∈ R and λ > 0. This looks like two exponentials
symmetric on both sides of the yaxis. Which of the methods below
would be very reasonable to use to generate realizations from this
distribution?
Answer: Inverse Transform Method AND Acceptance-Rejection
◉ Consider a bivariate normal random variable (X, Y ), for which
E[X] = −3, Var(X) = 4, E[Y ] = −2, Var(Y ) = 9, and Cov(X, Y ) = 2. Find
the Cholesky matrix associated with (X, Y ), i.e., the lower-triangular
matrix C such that Σ = CC0 , where Σ is the variance-covariance
matrix.
Answer: C = (2 0
1 2sqrt(2))
,◉ Consider a nonhomogeneous Poisson arrival process with rate
function λ(t) = 2t for t ≥ 0. Find the probability that there will be
exactly 2 arrivals between times t = 1 and 2.
Answer: 0.224
◉ Suppose we are generating arrivals from a nonhomogeneous
Poisson process with rate function λ(t) = 1 + sin(πt), so that the
maximum rate is λ ? = 2, which is periodically achieved. Suppose that
we generate a potential arrival (i.e., one at rate λ ? ) at time t = 0.75.
What is the probability that our usual thinning algorithm will
actually accept that potential arrival as an actual arrival? (Note that
the π means that calculations are in radians.)
Answer: 0.854
◉ Suppose X1, X2, . . . is an i.i.d. sequence of random variables with
mean µ and variance σ 2 . Consider the process Yn(t) ≡ Pbntc i=1 (Xi
− µ)/(σ √ n) for t ≥ 0. What is the asymptotic probability that Yn(4)
will be at least 2 as n becomes large? Hint: Recall that Donsker's
Theorem states that Yn(t) converges to a standard Brownian motion
as n becomes large.
Answer: 0.1587
◉ Which one of the following properties of a Brownian motion
process W(t) is FALSE?
, Answer: W(3) − W(1) is independent of W(4) − W(2).
◉ Find the sample variance of −10, 10, 0.
Answer: 100
S^2 = 100
◉ If X1, . . . , X10 are i.i.d. Exp(1/7) (i.e., having mean 7), what is the
expected value of the sample variance S 2 ?
Answer: 49
S^2 is always unbiased for the variance of Xi. Thus, we have E[S^2] =
Var(Xi) = 1/lambda^2 = 49.
◉ TRUE or FALSE? The mean squared error of an estimator is the
square of the bias plus the square of its variance
Answer: False
◉ If X1 = 7, X2 = 3, and X3 = 5 are i.i.d. realizations from a Nor(µ, σ2
) distribution, what is the value of the maximum likelihood estimate
for the variance σ 2 ?
Answer: 2.667
◉ Suppose that we take three i.i.d. observations X1 = 2, X2 = 3, and
X3 = 1 from X ∼ Exp(λ). Using the maximum likelihood estimate for
λ, find the MLE of Pr(X > 2).
COMPREHENSIVE SCRIPT WITH SOLUTIONS
◉ Suppose that Z1, Z2, and Z3 are i.i.d. Nor(0,1) random variables,
and let T = Z1 /sqrt((Z 2 2 + Z 2 3 )/2) . Find the value of x such that
Pr(T < x) = 0.99.
Answer: x=6.965
◉ Suppose X has the Laplace distribution with p.d.f. f(x) = λ/2
e^−λ|x| for x ∈ R and λ > 0. This looks like two exponentials
symmetric on both sides of the yaxis. Which of the methods below
would be very reasonable to use to generate realizations from this
distribution?
Answer: Inverse Transform Method AND Acceptance-Rejection
◉ Consider a bivariate normal random variable (X, Y ), for which
E[X] = −3, Var(X) = 4, E[Y ] = −2, Var(Y ) = 9, and Cov(X, Y ) = 2. Find
the Cholesky matrix associated with (X, Y ), i.e., the lower-triangular
matrix C such that Σ = CC0 , where Σ is the variance-covariance
matrix.
Answer: C = (2 0
1 2sqrt(2))
,◉ Consider a nonhomogeneous Poisson arrival process with rate
function λ(t) = 2t for t ≥ 0. Find the probability that there will be
exactly 2 arrivals between times t = 1 and 2.
Answer: 0.224
◉ Suppose we are generating arrivals from a nonhomogeneous
Poisson process with rate function λ(t) = 1 + sin(πt), so that the
maximum rate is λ ? = 2, which is periodically achieved. Suppose that
we generate a potential arrival (i.e., one at rate λ ? ) at time t = 0.75.
What is the probability that our usual thinning algorithm will
actually accept that potential arrival as an actual arrival? (Note that
the π means that calculations are in radians.)
Answer: 0.854
◉ Suppose X1, X2, . . . is an i.i.d. sequence of random variables with
mean µ and variance σ 2 . Consider the process Yn(t) ≡ Pbntc i=1 (Xi
− µ)/(σ √ n) for t ≥ 0. What is the asymptotic probability that Yn(4)
will be at least 2 as n becomes large? Hint: Recall that Donsker's
Theorem states that Yn(t) converges to a standard Brownian motion
as n becomes large.
Answer: 0.1587
◉ Which one of the following properties of a Brownian motion
process W(t) is FALSE?
, Answer: W(3) − W(1) is independent of W(4) − W(2).
◉ Find the sample variance of −10, 10, 0.
Answer: 100
S^2 = 100
◉ If X1, . . . , X10 are i.i.d. Exp(1/7) (i.e., having mean 7), what is the
expected value of the sample variance S 2 ?
Answer: 49
S^2 is always unbiased for the variance of Xi. Thus, we have E[S^2] =
Var(Xi) = 1/lambda^2 = 49.
◉ TRUE or FALSE? The mean squared error of an estimator is the
square of the bias plus the square of its variance
Answer: False
◉ If X1 = 7, X2 = 3, and X3 = 5 are i.i.d. realizations from a Nor(µ, σ2
) distribution, what is the value of the maximum likelihood estimate
for the variance σ 2 ?
Answer: 2.667
◉ Suppose that we take three i.i.d. observations X1 = 2, X2 = 3, and
X3 = 1 from X ∼ Exp(λ). Using the maximum likelihood estimate for
λ, find the MLE of Pr(X > 2).
