1|Page
ISYE 6644 SIMULATION & MODELLING FOR
ENGINEERING & SCIENCE - ISYE 6644 EXAMS
BUNDLE LATEST // ISYE 6644 FINAL AND MIDTERM
EXAM (NEW!)
(8.3) Find the sample variance of -3, -2, -1, 0, 1, 2, 3 -
...ANSWER...✓✓ 14/3 (or 4.666). If sample is entire
population than variance is 4.
(8.1) M/M/1 queue - ...ANSWER...✓✓ queue length having a
single server.
(8.3) If the expected value of your estimator equals the
parameter that you're trying to estimate, then your
estimator is unbiased. True of False - ...ANSWER...✓✓
True. This is the definition of unbiasedness
(8.3) If X1, X2, ..., Xn are i.i.d. with mean mu, then the
sample mean X-bar is unbiased for mu. True or False -
...ANSWER...✓✓ True.
(8.4) What is the MSE (Mean Squared Error) of an
estimator? - ...ANSWER...✓✓ Bias^2 + Variance
,2|Page
(8.3) What is the expected value of the mean of a Pois(λ)
random variable? - ...ANSWER...✓✓ λ is the mean and the
variance
(8.3) What is the expected sample variance s^2 of a
Pois(λ) random variable? - ...ANSWER...✓✓ λ is the
sample variance and the mean
(8.4) Suppose that estimator A has bias = 3 and variance =
12, while estimator B has bias -2 and variance = 14. Which
estimator (A or B) has the lower mean squared error? -
...ANSWER...✓✓ B is lower. Bias^2 + Variance: 18 < 21
MLE - ...ANSWER...✓✓ Maximum Likelihood Estimator -
"A method of estimating the parameters of a distribution
by maximizing a likelihood function, so that under the
assumed statistical model the observed data is most
probable."
(8.4) Suppose that X1=4, X2=3, X3=5 are i.i.d. realizations
from an Exp(λ) distribution. What is the MLE of λ? -
...ANSWER...✓✓ 0.25
, 3|Page
(8.5/8.6) If X1=2, X2=−2, and X3=0 are i.i.d. realizations
from a Nor(μ , σ^2) distribution, what is the value of the
maximum likelihood estimate for the variance σ^2? -
...ANSWER...✓✓ 8/3. MLE of σ^2 is the summation of the
squared differences (Xi - μ), all divided by n.
(8.5/8.6) Suppose we observe the Pois(λ) realizations
X1=5, X2=9 and X3=1. What is the maximum likelihood
estimate of λ? - ...ANSWER...✓✓ 5. λ is estimated as the
summation of sample values divided by the number of
sample values. (5+9+1)/3 = 5
(8.5) Suppose X1, ..., Xn are i.i.d. Bern(p). Find the MLE for
p. - ...ANSWER...✓✓
(8.7) Suppose that we have a number of observations
from a Pois(λ) distribution, and it turns out that the MLE
for λ is λhat=5. What's the maximum likelihood estimate
of Pr(X=3)? - ...ANSWER...✓✓ 0.1404. P(X=x) = λ^x * e^(−λ)
/ x!
ISYE 6644 SIMULATION & MODELLING FOR
ENGINEERING & SCIENCE - ISYE 6644 EXAMS
BUNDLE LATEST // ISYE 6644 FINAL AND MIDTERM
EXAM (NEW!)
(8.3) Find the sample variance of -3, -2, -1, 0, 1, 2, 3 -
...ANSWER...✓✓ 14/3 (or 4.666). If sample is entire
population than variance is 4.
(8.1) M/M/1 queue - ...ANSWER...✓✓ queue length having a
single server.
(8.3) If the expected value of your estimator equals the
parameter that you're trying to estimate, then your
estimator is unbiased. True of False - ...ANSWER...✓✓
True. This is the definition of unbiasedness
(8.3) If X1, X2, ..., Xn are i.i.d. with mean mu, then the
sample mean X-bar is unbiased for mu. True or False -
...ANSWER...✓✓ True.
(8.4) What is the MSE (Mean Squared Error) of an
estimator? - ...ANSWER...✓✓ Bias^2 + Variance
,2|Page
(8.3) What is the expected value of the mean of a Pois(λ)
random variable? - ...ANSWER...✓✓ λ is the mean and the
variance
(8.3) What is the expected sample variance s^2 of a
Pois(λ) random variable? - ...ANSWER...✓✓ λ is the
sample variance and the mean
(8.4) Suppose that estimator A has bias = 3 and variance =
12, while estimator B has bias -2 and variance = 14. Which
estimator (A or B) has the lower mean squared error? -
...ANSWER...✓✓ B is lower. Bias^2 + Variance: 18 < 21
MLE - ...ANSWER...✓✓ Maximum Likelihood Estimator -
"A method of estimating the parameters of a distribution
by maximizing a likelihood function, so that under the
assumed statistical model the observed data is most
probable."
(8.4) Suppose that X1=4, X2=3, X3=5 are i.i.d. realizations
from an Exp(λ) distribution. What is the MLE of λ? -
...ANSWER...✓✓ 0.25
, 3|Page
(8.5/8.6) If X1=2, X2=−2, and X3=0 are i.i.d. realizations
from a Nor(μ , σ^2) distribution, what is the value of the
maximum likelihood estimate for the variance σ^2? -
...ANSWER...✓✓ 8/3. MLE of σ^2 is the summation of the
squared differences (Xi - μ), all divided by n.
(8.5/8.6) Suppose we observe the Pois(λ) realizations
X1=5, X2=9 and X3=1. What is the maximum likelihood
estimate of λ? - ...ANSWER...✓✓ 5. λ is estimated as the
summation of sample values divided by the number of
sample values. (5+9+1)/3 = 5
(8.5) Suppose X1, ..., Xn are i.i.d. Bern(p). Find the MLE for
p. - ...ANSWER...✓✓
(8.7) Suppose that we have a number of observations
from a Pois(λ) distribution, and it turns out that the MLE
for λ is λhat=5. What's the maximum likelihood estimate
of Pr(X=3)? - ...ANSWER...✓✓ 0.1404. P(X=x) = λ^x * e^(−λ)
/ x!
