ISYE 6644 COMPREHENSIVE EXAM
QUESTIONS AND CORRECT ANSWERS
GRADED A+
◉ (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
◉ (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
◉ (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.
QUESTIONS AND CORRECT ANSWERS
GRADED A+
◉ (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
◉ (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
◉ (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.
