©SIRJOEL EXAM SOLUTIONS 2024/2025
ALL RIGHTS RESERVED.
ISYE 6644 Simulation Exam Questions And
Answers
(8.3) Find the sample variance of -3, -2, -1, 0, 1, 2, 3 - Answers✔14/3 (or 4.666). If sample is
entire population than variance is 4.
(8.1) M/M/1 queue - Answers✔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 - Answers✔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 - Answers✔True.
(8.4) What is the MSE (Mean Squared Error) of an estimator? - Answers✔Bias^2 + Variance
(8.3) What is the expected value of the mean of a Pois(λ) random variable? - Answers✔λ is the
mean and the variance
(8.3) What is the expected sample variance s^2 of a Pois(λ) random variable? - Answers✔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? - Answers✔B is
lower. Bias^2 + Variance: 18 < 21
MLE - Answers✔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? - Answers✔0.25
, ©SIRJOEL EXAM SOLUTIONS 2024/2025
ALL RIGHTS RESERVED.
(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? - Answers✔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? - Answers✔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. - Answers✔
(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)? -
Answers✔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. - Answers✔True
(8.6) TRUE or FALSE? Sometimes it might be difficult to obtain an MLE in closed form. -
Answers✔True. (There is a gamma example.)
(8.7) Suppose that the MLE for a parameter θ is θhat=4. Find the MLE for √θ. - Answers✔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]? - Answers✔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? - Answers✔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? - Answers✔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? -
Answers✔Reject. The test statistic 9.12 is not less than 5.99.
ALL RIGHTS RESERVED.
ISYE 6644 Simulation Exam Questions And
Answers
(8.3) Find the sample variance of -3, -2, -1, 0, 1, 2, 3 - Answers✔14/3 (or 4.666). If sample is
entire population than variance is 4.
(8.1) M/M/1 queue - Answers✔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 - Answers✔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 - Answers✔True.
(8.4) What is the MSE (Mean Squared Error) of an estimator? - Answers✔Bias^2 + Variance
(8.3) What is the expected value of the mean of a Pois(λ) random variable? - Answers✔λ is the
mean and the variance
(8.3) What is the expected sample variance s^2 of a Pois(λ) random variable? - Answers✔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? - Answers✔B is
lower. Bias^2 + Variance: 18 < 21
MLE - Answers✔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? - Answers✔0.25
, ©SIRJOEL EXAM SOLUTIONS 2024/2025
ALL RIGHTS RESERVED.
(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? - Answers✔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? - Answers✔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. - Answers✔
(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)? -
Answers✔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. - Answers✔True
(8.6) TRUE or FALSE? Sometimes it might be difficult to obtain an MLE in closed form. -
Answers✔True. (There is a gamma example.)
(8.7) Suppose that the MLE for a parameter θ is θhat=4. Find the MLE for √θ. - Answers✔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]? - Answers✔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? - Answers✔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? - Answers✔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? -
Answers✔Reject. The test statistic 9.12 is not less than 5.99.
