ISIYE 6644 SIMULATION EXAM QUESTIONS WITH
CORRECT ANSWERS THAT ARE ALREADY GRADED
WITH A 100% GUARANTEED PASS
(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. - ✔✔True
(8.6) TRUE or FALSE? Sometimes it might be difficult to obtain an MLE in closed form. - ✔✔True. (There
is a gamma example.)
(8.7) Suppose that the MLE for a parameter θ is θhat=4. Find the MLE for √θ. - ✔✔2. 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]? - ✔✔35.6667. 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? -
✔✔Reject. 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.
, (8.9) Suppose H0 is true, but you've just rejected it! What have you done? - ✔✔Type I error
(8.10/8.11) The test statistic is χ0^2 = 9.12. Now, let's use our old friend α = 0.05 in our test. Let k = 4
denote the number of cells (that we ultimately ended up with) and let s = 1 denote the number of
parameters we had to estimate. Then we compare against χ^2(α=0.05 , k − s − 1) = χ^2(α=0.05 , 2) =
5.99. Do we ACCEPT (i.e., fail to reject) or REJECT the Geometric hypothesis? - ✔✔Reject. The test
statistic 9.12 is not less than 5.99.
(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?
- ✔✔Accept. 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! - ✔✔True
(9.1) We often distinguish between two general types of simulations with regard to output analysis.
What are they called? - ✔✔Finite-horizon and steady-state
What are i.i.d. random variables? - ✔✔It 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".
(9.2) TRUE or FALSE? Suppose that X1,X2,...,Xn are consecutive waiting times, and we define the sample
mean X¯=∑Xi/n. Then Var(X¯)=Var(Xi)/n. - ✔✔False. Very FALSE! (The issue is that correlation between
the observations messes up the variance of the sample mean. In fact, this is one of the main reasons
why output analysis is difficult!)
(9.4) TRUE or FALSE? You can also conduct finite-horizon estimation for quantities other than expected
values, e.g., simulate a bank from 8:00 a.m. to 5:00 p.m., and find a confidence interval for the 95th
quantile of customer waiting times. - ✔✔True
CORRECT ANSWERS THAT ARE ALREADY GRADED
WITH A 100% GUARANTEED PASS
(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. - ✔✔True
(8.6) TRUE or FALSE? Sometimes it might be difficult to obtain an MLE in closed form. - ✔✔True. (There
is a gamma example.)
(8.7) Suppose that the MLE for a parameter θ is θhat=4. Find the MLE for √θ. - ✔✔2. 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]? - ✔✔35.6667. 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? -
✔✔Reject. 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.
, (8.9) Suppose H0 is true, but you've just rejected it! What have you done? - ✔✔Type I error
(8.10/8.11) The test statistic is χ0^2 = 9.12. Now, let's use our old friend α = 0.05 in our test. Let k = 4
denote the number of cells (that we ultimately ended up with) and let s = 1 denote the number of
parameters we had to estimate. Then we compare against χ^2(α=0.05 , k − s − 1) = χ^2(α=0.05 , 2) =
5.99. Do we ACCEPT (i.e., fail to reject) or REJECT the Geometric hypothesis? - ✔✔Reject. The test
statistic 9.12 is not less than 5.99.
(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?
- ✔✔Accept. 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! - ✔✔True
(9.1) We often distinguish between two general types of simulations with regard to output analysis.
What are they called? - ✔✔Finite-horizon and steady-state
What are i.i.d. random variables? - ✔✔It 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".
(9.2) TRUE or FALSE? Suppose that X1,X2,...,Xn are consecutive waiting times, and we define the sample
mean X¯=∑Xi/n. Then Var(X¯)=Var(Xi)/n. - ✔✔False. Very FALSE! (The issue is that correlation between
the observations messes up the variance of the sample mean. In fact, this is one of the main reasons
why output analysis is difficult!)
(9.4) TRUE or FALSE? You can also conduct finite-horizon estimation for quantities other than expected
values, e.g., simulate a bank from 8:00 a.m. to 5:00 p.m., and find a confidence interval for the 95th
quantile of customer waiting times. - ✔✔True
