ISYE-6644 Simulation Questions And Verified Detailed Answers
(8.3) Find the sample variance of -3, -2, -1, 0, 1, 2, 3 - CORRECT ANSWER✔14/3 (or 4.666). If
sample is entire population than variance is 4.
(8.1) M/M/1 queue - CORRECT 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 - CORRECT 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 - CORRECT ANSWER✔True.
(8.4) What is the MSE (Mean Squared Error) of an estimator? - CORRECT ANSWER✔Bias^2 +
Variance
(8.3) What is the expected value of the mean of a Pois(λ) random variable? - CORRECT
ANSWER✔λ is the mean and the variance
(8.3) What is the expected sample variance s^2 of a Pois(λ) random variable? - CORRECT
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? - CORRECT
ANSWER✔B is lower. Bias^2 + Variance: 18 < 21
, MLE - CORRECT 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? - CORRECT 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? - CORRECT 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? - CORRECT 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. - CORRECT 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)? -
CORRECT ANSWER✔0.1404. P(X=x) = λ^x * e^(−λ) / x!
(8.6) TRUE or FALSE? It's possible to estimate two MLEs simultaneously, e.g., for the Nor(μ,σ2)
distribution. - CORRECT ANSWER✔True
(8.6) TRUE or FALSE? Sometimes it might be difficult to obtain an MLE in closed form. -
CORRECT ANSWER✔True. (There is a gamma example.)
(8.7) Suppose that the MLE for a parameter θ is θhat=4. Find the MLE for √θ. - CORRECT
ANSWER✔2. Invariance immediately implies that the MLE of √θ is simply √θhat = 2
(8.3) Find the sample variance of -3, -2, -1, 0, 1, 2, 3 - CORRECT ANSWER✔14/3 (or 4.666). If
sample is entire population than variance is 4.
(8.1) M/M/1 queue - CORRECT 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 - CORRECT 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 - CORRECT ANSWER✔True.
(8.4) What is the MSE (Mean Squared Error) of an estimator? - CORRECT ANSWER✔Bias^2 +
Variance
(8.3) What is the expected value of the mean of a Pois(λ) random variable? - CORRECT
ANSWER✔λ is the mean and the variance
(8.3) What is the expected sample variance s^2 of a Pois(λ) random variable? - CORRECT
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? - CORRECT
ANSWER✔B is lower. Bias^2 + Variance: 18 < 21
, MLE - CORRECT 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? - CORRECT 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? - CORRECT 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? - CORRECT 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. - CORRECT 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)? -
CORRECT ANSWER✔0.1404. P(X=x) = λ^x * e^(−λ) / x!
(8.6) TRUE or FALSE? It's possible to estimate two MLEs simultaneously, e.g., for the Nor(μ,σ2)
distribution. - CORRECT ANSWER✔True
(8.6) TRUE or FALSE? Sometimes it might be difficult to obtain an MLE in closed form. -
CORRECT ANSWER✔True. (There is a gamma example.)
(8.7) Suppose that the MLE for a parameter θ is θhat=4. Find the MLE for √θ. - CORRECT
ANSWER✔2. Invariance immediately implies that the MLE of √θ is simply √θhat = 2
