ISYE 6644 MIDTERM 2 NEWEST
CERTIFICATION REVIEW SET 2026
ANSWERS GUARANTEED PASS
⫸ (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.
⫸ (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:
CERTIFICATION REVIEW SET 2026
ANSWERS GUARANTEED PASS
⫸ (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.
⫸ (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:
