ISYE-6644 SIMULATION QUESTIONS
& ANSWERS 100% CORRECT!!
(8.3) Find the sample variance of -3, -2, -1, 0, 1, 2, 3 - ANSWER14/3 (or 4.666). If
sample is entire population than variance is 4.
(8.1) M/M/1 queue - ANSWERqueue 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 - ANSWERTrue. 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 - ANSWERTrue.
(8.4) What is the MSE (Mean Squared Error) of an estimator? - ANSWERBias^2 +
Variance
(8.3) What is the expected value of the mean of a Pois(λ) random variable? -
ANSWERλ is the mean and the variance
(8.3) What is the expected sample variance s^2 of a Pois(λ) random variable? -
ANSWERλ 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? - ANSWERB is lower. Bias^2 + Variance: 18 < 21
(8.12) Consider the PRN's U1 = 0.1 , U2 = 0.9 , and U3 = 0.2. Use Kolmogorov-
Smirnov with α = 0.05 to test to see if these numbers are indeed uniform. Do we
ACCEPT (i.e., fail to reject) or REJECT uniformity? - ANSWERAccept. From table,
D(α=0.05, 3) = 0.70760. Create ordered sample set: 0.1, 0.2, 0.9. Since the max
value of D test is 0.467, then we fail to reject because it is smaller.
(9.1) TRUE or FALSE? Simulation output (e.g., consecutive customer waiting times)
is almost never i.i.d. normal - and that's a big fat problem! - ANSWERTrue
(9.1) We often distinguish between two general types of simulations with regard to
output analysis. What are they called? - ANSWERFinite-horizon and steady-state
What are i.i.d. random variables? - ANSWERIt means "Independent and identically
distributed".
A good example is a succession of throws of a fair coin: The coin has no memory, so
all the throws are "independent".
, And every throw is 50:50 (heads:tails), so the coin is and stays fair - the distribution
from which every throw is drawn, so to speak, is and stays the same: "identically
distributed".
MLE - ANSWERMaximum 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? - ANSWER0.25
(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? - ANSWER8/3. 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? - ANSWER5. 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)? - ANSWER0.1404. 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. - ANSWERTrue
(8.6) TRUE or FALSE? Sometimes it might be difficult to obtain an MLE in closed
form. - ANSWERTrue. (There is a gamma example.)
(8.7) Suppose that the MLE for a parameter θ is θhat=4. Find the MLE for √θ. -
ANSWER2. Invariance immediately implies that the MLE of √θ is simply √θhat = 2
(8.8) Suppose that we observe X1 = 5, X2 = 9, and X3 = 1. What's the method of
moments estimate of E[X^2]? - ANSWER35.6667. Second moment is the sum of the
squared samples divided by the number of samples. (5^2 + 9^2 + 1^2) / 3 =
35.666666667
(8.9) Suppose we're conducting a χ^2 goodness-of-fit test with Type I error rate α =
0.01 to determine whether or not 100 i.i.d. observations are from a lognormal
distribution with unknown parameters μ and σ^2. If we divide the observations into 5
equal-probability intervals and we observe a g-o-f statistic of χ0^2 = 11.2, will we
ACCEPT (i.e., fail to reject) or REJECT the null hypothesis of lognormality? -
ANSWERReject. k = 5, subtract 1 and subtract 2 for the two unknown parameters (or
had to estimate), so degrees of freedom is 2. critical value for dof 2 and alpha 0.01 is
9.21. 11.2 is not smaller than 9.21 so we reject it. Not a good fit.
& ANSWERS 100% CORRECT!!
(8.3) Find the sample variance of -3, -2, -1, 0, 1, 2, 3 - ANSWER14/3 (or 4.666). If
sample is entire population than variance is 4.
(8.1) M/M/1 queue - ANSWERqueue 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 - ANSWERTrue. 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 - ANSWERTrue.
(8.4) What is the MSE (Mean Squared Error) of an estimator? - ANSWERBias^2 +
Variance
(8.3) What is the expected value of the mean of a Pois(λ) random variable? -
ANSWERλ is the mean and the variance
(8.3) What is the expected sample variance s^2 of a Pois(λ) random variable? -
ANSWERλ 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? - ANSWERB is lower. Bias^2 + Variance: 18 < 21
(8.12) Consider the PRN's U1 = 0.1 , U2 = 0.9 , and U3 = 0.2. Use Kolmogorov-
Smirnov with α = 0.05 to test to see if these numbers are indeed uniform. Do we
ACCEPT (i.e., fail to reject) or REJECT uniformity? - ANSWERAccept. From table,
D(α=0.05, 3) = 0.70760. Create ordered sample set: 0.1, 0.2, 0.9. Since the max
value of D test is 0.467, then we fail to reject because it is smaller.
(9.1) TRUE or FALSE? Simulation output (e.g., consecutive customer waiting times)
is almost never i.i.d. normal - and that's a big fat problem! - ANSWERTrue
(9.1) We often distinguish between two general types of simulations with regard to
output analysis. What are they called? - ANSWERFinite-horizon and steady-state
What are i.i.d. random variables? - ANSWERIt means "Independent and identically
distributed".
A good example is a succession of throws of a fair coin: The coin has no memory, so
all the throws are "independent".
, And every throw is 50:50 (heads:tails), so the coin is and stays fair - the distribution
from which every throw is drawn, so to speak, is and stays the same: "identically
distributed".
MLE - ANSWERMaximum 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? - ANSWER0.25
(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? - ANSWER8/3. 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? - ANSWER5. 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)? - ANSWER0.1404. 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. - ANSWERTrue
(8.6) TRUE or FALSE? Sometimes it might be difficult to obtain an MLE in closed
form. - ANSWERTrue. (There is a gamma example.)
(8.7) Suppose that the MLE for a parameter θ is θhat=4. Find the MLE for √θ. -
ANSWER2. Invariance immediately implies that the MLE of √θ is simply √θhat = 2
(8.8) Suppose that we observe X1 = 5, X2 = 9, and X3 = 1. What's the method of
moments estimate of E[X^2]? - ANSWER35.6667. Second moment is the sum of the
squared samples divided by the number of samples. (5^2 + 9^2 + 1^2) / 3 =
35.666666667
(8.9) Suppose we're conducting a χ^2 goodness-of-fit test with Type I error rate α =
0.01 to determine whether or not 100 i.i.d. observations are from a lognormal
distribution with unknown parameters μ and σ^2. If we divide the observations into 5
equal-probability intervals and we observe a g-o-f statistic of χ0^2 = 11.2, will we
ACCEPT (i.e., fail to reject) or REJECT the null hypothesis of lognormality? -
ANSWERReject. k = 5, subtract 1 and subtract 2 for the two unknown parameters (or
had to estimate), so degrees of freedom is 2. critical value for dof 2 and alpha 0.01 is
9.21. 11.2 is not smaller than 9.21 so we reject it. Not a good fit.
