ISYE-6644 Simulation Questions with Verified
Correct Answers
(8.3) Find the sample variance of -3, -2, -1, 0, 1, 2, 3
14/3 (or 4.666). If sample is entire population than variance is 4.
(8.1) M/M/1 queue
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
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
True.
(8.4) What is the MSE (Mean Squared Error) of an estimator?
Bias^2 + Variance
(8.3) What is the expected value of the mean of a Pois(λ) random variable?
λ is the mean and the variance
(8.3) What is the expected sample variance s^2 of a Pois(λ) random variable?
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?
B is lower. Bias^2 + Variance: 18 < 21
, MLE
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?
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?
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?
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.
(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)?
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 Answers
(8.3) Find the sample variance of -3, -2, -1, 0, 1, 2, 3
14/3 (or 4.666). If sample is entire population than variance is 4.
(8.1) M/M/1 queue
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
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
True.
(8.4) What is the MSE (Mean Squared Error) of an estimator?
Bias^2 + Variance
(8.3) What is the expected value of the mean of a Pois(λ) random variable?
λ is the mean and the variance
(8.3) What is the expected sample variance s^2 of a Pois(λ) random variable?
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?
B is lower. Bias^2 + Variance: 18 < 21
, MLE
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?
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?
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?
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.
(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)?
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.
