ISYE-6644 SIMULATION Questions with Correct Solutions
Question:
1. (8.3) Find the sample variance of -3, -2, -1, 0, 1, 2, 3
Answer:
14/3 (or 4.666). If sample is entire population than variance is 4.
Question:
2. (8.1) M/M/1 queue
Answer:
queue length having a single server.
Question:
3. (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
Answer:
True. This is the definition of unbiasedness
Question:
4. (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
Answer:
True.
Question:
5. (8.4) What is the MSE (Mean Squared Error) of an estimator?
Answer:
Bias^2 + Variance
Question:
6. (8.3) What is the expected value of the mean of a Pois(λ) random variable?
Answer:
λ is the mean and the variance
Question:
7. (8.3) What is the expected sample variance s^2 of a Pois(λ) random variable?
, Answer:
λ is the sample variance and the mean
Question:
8. (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?
Answer:
B is lower. Bias^2 + Variance: 18 < 21
Question:
9. MLE
Answer:
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."
Question:
10. (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?
Answer:
0.25
Question:
11. (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?
Answer:
8/3. MLE of σ^2 is the summation of the squared differences (Xi - μ), all divided by n.
Question:
12. (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?
Answer:
5. λ is estimated as the summation of sample values divided by the number of sample values.
(5+9+1)/3 = 5
Question:
13. (8.5) Suppose X1, ..., Xn are i.i.d. Bern(p). Find the MLE for p.
Question:
1. (8.3) Find the sample variance of -3, -2, -1, 0, 1, 2, 3
Answer:
14/3 (or 4.666). If sample is entire population than variance is 4.
Question:
2. (8.1) M/M/1 queue
Answer:
queue length having a single server.
Question:
3. (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
Answer:
True. This is the definition of unbiasedness
Question:
4. (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
Answer:
True.
Question:
5. (8.4) What is the MSE (Mean Squared Error) of an estimator?
Answer:
Bias^2 + Variance
Question:
6. (8.3) What is the expected value of the mean of a Pois(λ) random variable?
Answer:
λ is the mean and the variance
Question:
7. (8.3) What is the expected sample variance s^2 of a Pois(λ) random variable?
, Answer:
λ is the sample variance and the mean
Question:
8. (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?
Answer:
B is lower. Bias^2 + Variance: 18 < 21
Question:
9. MLE
Answer:
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."
Question:
10. (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?
Answer:
0.25
Question:
11. (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?
Answer:
8/3. MLE of σ^2 is the summation of the squared differences (Xi - μ), all divided by n.
Question:
12. (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?
Answer:
5. λ is estimated as the summation of sample values divided by the number of sample values.
(5+9+1)/3 = 5
Question:
13. (8.5) Suppose X1, ..., Xn are i.i.d. Bern(p). Find the MLE for p.
