ISYE-6644 Simulation
• (10.1) Which of the following parameters can you get confidence intervals for?
Means, Variances, Quantiles, Differences between the means of two systems, or
all of those. - ✔️✔️All. We can get CIs for means, variances, quantiles, and
differences between the means of two systems.
• (Lesson 8.2: Identifying Distributions.) Let's play Name That Distribution!
• The number of times a "3" comes up in 10 dice tosses.
• Bernoulli
• Binomial
• Geometric
• Negative Binomial
• Pareto - ✔️✔️b. Binomial
• (9.7) True or False. The method of batch means is easy to use. - ✔️✔️True
• (9.7) True or False. Batch means chops the consecutive observations into a
number of nonoverlapping, contiguous batches. - ✔️✔️True
• (9.7) True or False. You can use the method of batch means to obtain a
confidence interval for the steady-state mean μ. - ✔️✔️True
• (9.7) True or False. The batch means estimator for the variance parameter σ^2 is
asymptotically unbiased as the batch size m→∞. - ✔️✔️True
• Bernoulli probability selection problem - ✔️✔️Bunch of Bernoulli populations and
find the one with the best success probability
• Multinomial cell selection problem - ✔️✔️
• Normal means ranking and selection problem - ✔️✔️Bunch of normal
distributions and we want to find the one with the largest or smallest mean.
, • (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
• (10.1) Which of the following parameters can you get confidence intervals for?
Means, Variances, Quantiles, Differences between the means of two systems, or
all of those. - ✔️✔️All. We can get CIs for means, variances, quantiles, and
differences between the means of two systems.
• (Lesson 8.2: Identifying Distributions.) Let's play Name That Distribution!
• The number of times a "3" comes up in 10 dice tosses.
• Bernoulli
• Binomial
• Geometric
• Negative Binomial
• Pareto - ✔️✔️b. Binomial
• (9.7) True or False. The method of batch means is easy to use. - ✔️✔️True
• (9.7) True or False. Batch means chops the consecutive observations into a
number of nonoverlapping, contiguous batches. - ✔️✔️True
• (9.7) True or False. You can use the method of batch means to obtain a
confidence interval for the steady-state mean μ. - ✔️✔️True
• (9.7) True or False. The batch means estimator for the variance parameter σ^2 is
asymptotically unbiased as the batch size m→∞. - ✔️✔️True
• Bernoulli probability selection problem - ✔️✔️Bunch of Bernoulli populations and
find the one with the best success probability
• Multinomial cell selection problem - ✔️✔️
• Normal means ranking and selection problem - ✔️✔️Bunch of normal
distributions and we want to find the one with the largest or smallest mean.
, • (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
