ISYE-6644 Simulation
Study online at https://quizlet.com/_8a2u5u
1. (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.
2. (8.1) M/M/1 queue queue length having a sin-
gle server.
3. (8.3) If the expected value of your estimator equals True. This is the definition
the parameter that you're trying to estimate, then your of unbiasedness
estimator is unbiased. True of False
4. (8.3) If X1, X2, ..., Xn are i.i.d. with mean mu, then the True.
sample mean X-bar is unbiased for mu. True or False
5. (8.4) What is the MSE (Mean Squared Error) of an esti- Bias^2 + Variance
mator?
6. (8.3) What is the expected value of the mean of a Pois(λ) λ is the mean and the vari-
random variable? ance
7. (8.3) What is the expected sample variance s^2 of a λ is the sample variance
Pois(λ) random variable? and the mean
8. (8.4) Suppose that estimator A has bias = 3 and vari- B is lower. Bias^2 + Vari-
ance = 12, while estimator B has bias -2 and variance ance: 18 < 21
= 14. Which estimator (A or B) has the lower mean
squared error?
9. MLE Maximum Likelihood Esti-
mator - "A method of esti-
mating the parameters of
a distribution by maximiz-
ing a likelihood function,
, ISYE-6644 Simulation
Study online at https://quizlet.com/_8a2u5u
so that under the assumed
statistical model the ob-
served data is most prob-
able."
10. (8.4) Suppose that X1=4, X2=3, X3=5 are i.i.d. realiza- 0.25
tions from an Exp(λ) distribution. What is the MLE of λ?
11. (8.5/8.6) If X1=2, X2=−2, and X3=0 are i.i.d. realizations 8/3. MLE of σ^2 is the
from a Nor(μ , σ^2) distribution, what is the value of the summation of the squared
maximum likelihood estimate for the variance σ^2? differences (Xi - μ), all di-
vided by n.
12. (8.5/8.6) Suppose we observe the Pois(λ) realizations 5. λ is estimated as the
X1=5, X2=9 and X3=1. What is the maximum likelihood summation of sample val-
estimate of λ? ues divided by the num-
ber of sample values.
(5+9+1)/3 = 5
13. (8.5) Suppose X1, ..., Xn are i.i.d. Bern(p). Find the MLE
for p.
14. (8.7) Suppose that we have a number of observations 0.1404. P(X=x) = λ^x *
from a Pois(λ) distribution, and it turns out that the e^(−λ) / x!
MLE for λ is λhat=5. What's the maximum likelihood
estimate of Pr(X=3)?
15. (8.6) TRUE or FALSE? It's possible to estimate two MLEs True
simultaneously, e.g., for the Nor(μ,σ2) distribution.
16. (8.6) TRUE or FALSE? Sometimes it might be difficult to True. (There is a gamma
obtain an MLE in closed form. example.)
17.
Study online at https://quizlet.com/_8a2u5u
1. (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.
2. (8.1) M/M/1 queue queue length having a sin-
gle server.
3. (8.3) If the expected value of your estimator equals True. This is the definition
the parameter that you're trying to estimate, then your of unbiasedness
estimator is unbiased. True of False
4. (8.3) If X1, X2, ..., Xn are i.i.d. with mean mu, then the True.
sample mean X-bar is unbiased for mu. True or False
5. (8.4) What is the MSE (Mean Squared Error) of an esti- Bias^2 + Variance
mator?
6. (8.3) What is the expected value of the mean of a Pois(λ) λ is the mean and the vari-
random variable? ance
7. (8.3) What is the expected sample variance s^2 of a λ is the sample variance
Pois(λ) random variable? and the mean
8. (8.4) Suppose that estimator A has bias = 3 and vari- B is lower. Bias^2 + Vari-
ance = 12, while estimator B has bias -2 and variance ance: 18 < 21
= 14. Which estimator (A or B) has the lower mean
squared error?
9. MLE Maximum Likelihood Esti-
mator - "A method of esti-
mating the parameters of
a distribution by maximiz-
ing a likelihood function,
, ISYE-6644 Simulation
Study online at https://quizlet.com/_8a2u5u
so that under the assumed
statistical model the ob-
served data is most prob-
able."
10. (8.4) Suppose that X1=4, X2=3, X3=5 are i.i.d. realiza- 0.25
tions from an Exp(λ) distribution. What is the MLE of λ?
11. (8.5/8.6) If X1=2, X2=−2, and X3=0 are i.i.d. realizations 8/3. MLE of σ^2 is the
from a Nor(μ , σ^2) distribution, what is the value of the summation of the squared
maximum likelihood estimate for the variance σ^2? differences (Xi - μ), all di-
vided by n.
12. (8.5/8.6) Suppose we observe the Pois(λ) realizations 5. λ is estimated as the
X1=5, X2=9 and X3=1. What is the maximum likelihood summation of sample val-
estimate of λ? ues divided by the num-
ber of sample values.
(5+9+1)/3 = 5
13. (8.5) Suppose X1, ..., Xn are i.i.d. Bern(p). Find the MLE
for p.
14. (8.7) Suppose that we have a number of observations 0.1404. P(X=x) = λ^x *
from a Pois(λ) distribution, and it turns out that the e^(−λ) / x!
MLE for λ is λhat=5. What's the maximum likelihood
estimate of Pr(X=3)?
15. (8.6) TRUE or FALSE? It's possible to estimate two MLEs True
simultaneously, e.g., for the Nor(μ,σ2) distribution.
16. (8.6) TRUE or FALSE? Sometimes it might be difficult to True. (There is a gamma
obtain an MLE in closed form. example.)
17.
