ISYE 6644 COMPREHENSIVE TEST BANK
COMPLETE QUESTIONS AND ANSWERS
PRACTICE SOLUTION VERIFIED
◉ (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:
◉ (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!
COMPLETE QUESTIONS AND ANSWERS
PRACTICE SOLUTION VERIFIED
◉ (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:
◉ (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!
