ISYE-6644 Simulation Questions and Answers Best rated A+ Guaranteed
Success Latest Update ACTUAL UPDATED QUESTIONS AND CORRECT
ANSWERS
(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 True. This is the definition of unbiasedness
parameter that you're trying to estimate, then your
estimator is unbiased. True of False
(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
(8.4) What is the MSE (Mean Squared Error) of an Bias^2 + Variance
estimator?
(8.3) What is the expected value of the mean of a Pois(λ) λ is the mean and the variance
random variable?
(8.3) What is the expected sample variance s^2 of a λ is the sample variance and the mean
Pois(λ) random variable?
(8.4) Suppose that estimator A has bias = 3 and variance = B is lower. Bias^2 + Variance: 18 < 21
12, while estimator B has bias -2 and variance = 14. Which
estimator (A or B) has the lower mean squared error?
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 0.25
from an Exp(λ) distribution. What is the MLE of λ?
(8.5/8.6) If X1=2, X2=−2, and X3=0 are i.i.d. realizations from 8/3. MLE of σ^2 is the summation of the squared differences (Xi - μ), all divided by
a Nor(μ , σ^2) distribution, what is the value of the n.
maximum likelihood estimate for the variance σ^2?
(8.5/8.6) Suppose we observe the Pois(λ) realizations 5. λ is estimated as the summation of sample values divided by the number of
X1=5, X2=9 and X3=1. What is the maximum likelihood sample values. (5+9+1)/3 = 5
estimate of λ?
, (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 0.1404. P(X=x) = λ^x * e^(−λ) / x!
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)?
(8.6) TRUE or FALSE? It's possible to estimate two MLEs True
simultaneously, e.g., for the Nor(μ,σ2) distribution.
(8.6) TRUE or FALSE? Sometimes it might be difficult to True. (There is a gamma example.)
obtain an MLE in closed form.
(8.7) Suppose that the MLE for a parameter θ is θhat=4. 2. Invariance immediately implies that the MLE of √θ is simply √θhat = 2
Find the MLE for √θ.
(8.8) Suppose that we observe X1 = 5, X2 = 9, and X3 = 1. 35.6667. Second moment is the sum of the squared samples divided by the
What's the method of moments estimate of E[X^2]? number of samples. (5^2 + 9^2 + 1^2) / 3 = 35.666666667
(8.9) Suppose we're conducting a χ^2 goodness-of-fit Reject. k = 5, subtract 1 and subtract 2 for the two unknown parameters (or had to
test with Type I error rate α = 0.01 to determine whether estimate), so degrees of freedom is 2. critical value for dof 2 and alpha 0.01 is 9.21.
or not 100 i.i.d. observations are from a lognormal 11.2 is not smaller than 9.21 so we reject it. Not a good fit.
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?
(8.9) Suppose H0 is true, but you've just rejected it! What Type I error
have you done?
(8.10/8.11) The test statistic is χ0^2 = 9.12. Now, let's use our Reject. The test statistic 9.12 is not less than 5.99.
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?
(8.12) Consider the PRN's U1 = 0.1 , U2 = 0.9 , and U3 = 0.2. Accept. From table, D(α=0.05, 3) = 0.70760. Create ordered sample set: 0.1, 0.2, 0.9.
Use Kolmogorov-Smirnov with α = 0.05 to test to see if Since the max value of D test is 0.467, then we fail to reject because it is smaller.
these numbers are indeed uniform. Do we ACCEPT (i.e.,
fail to reject) or REJECT uniformity?
Success Latest Update ACTUAL UPDATED QUESTIONS AND CORRECT
ANSWERS
(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 True. This is the definition of unbiasedness
parameter that you're trying to estimate, then your
estimator is unbiased. True of False
(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
(8.4) What is the MSE (Mean Squared Error) of an Bias^2 + Variance
estimator?
(8.3) What is the expected value of the mean of a Pois(λ) λ is the mean and the variance
random variable?
(8.3) What is the expected sample variance s^2 of a λ is the sample variance and the mean
Pois(λ) random variable?
(8.4) Suppose that estimator A has bias = 3 and variance = B is lower. Bias^2 + Variance: 18 < 21
12, while estimator B has bias -2 and variance = 14. Which
estimator (A or B) has the lower mean squared error?
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 0.25
from an Exp(λ) distribution. What is the MLE of λ?
(8.5/8.6) If X1=2, X2=−2, and X3=0 are i.i.d. realizations from 8/3. MLE of σ^2 is the summation of the squared differences (Xi - μ), all divided by
a Nor(μ , σ^2) distribution, what is the value of the n.
maximum likelihood estimate for the variance σ^2?
(8.5/8.6) Suppose we observe the Pois(λ) realizations 5. λ is estimated as the summation of sample values divided by the number of
X1=5, X2=9 and X3=1. What is the maximum likelihood sample values. (5+9+1)/3 = 5
estimate of λ?
, (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 0.1404. P(X=x) = λ^x * e^(−λ) / x!
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)?
(8.6) TRUE or FALSE? It's possible to estimate two MLEs True
simultaneously, e.g., for the Nor(μ,σ2) distribution.
(8.6) TRUE or FALSE? Sometimes it might be difficult to True. (There is a gamma example.)
obtain an MLE in closed form.
(8.7) Suppose that the MLE for a parameter θ is θhat=4. 2. Invariance immediately implies that the MLE of √θ is simply √θhat = 2
Find the MLE for √θ.
(8.8) Suppose that we observe X1 = 5, X2 = 9, and X3 = 1. 35.6667. Second moment is the sum of the squared samples divided by the
What's the method of moments estimate of E[X^2]? number of samples. (5^2 + 9^2 + 1^2) / 3 = 35.666666667
(8.9) Suppose we're conducting a χ^2 goodness-of-fit Reject. k = 5, subtract 1 and subtract 2 for the two unknown parameters (or had to
test with Type I error rate α = 0.01 to determine whether estimate), so degrees of freedom is 2. critical value for dof 2 and alpha 0.01 is 9.21.
or not 100 i.i.d. observations are from a lognormal 11.2 is not smaller than 9.21 so we reject it. Not a good fit.
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?
(8.9) Suppose H0 is true, but you've just rejected it! What Type I error
have you done?
(8.10/8.11) The test statistic is χ0^2 = 9.12. Now, let's use our Reject. The test statistic 9.12 is not less than 5.99.
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?
(8.12) Consider the PRN's U1 = 0.1 , U2 = 0.9 , and U3 = 0.2. Accept. From table, D(α=0.05, 3) = 0.70760. Create ordered sample set: 0.1, 0.2, 0.9.
Use Kolmogorov-Smirnov with α = 0.05 to test to see if Since the max value of D test is 0.467, then we fail to reject because it is smaller.
these numbers are indeed uniform. Do we ACCEPT (i.e.,
fail to reject) or REJECT uniformity?
