Page 1 of 54
SIMULATION MODULE 1 ISYE 6644
PRACTICE EXAM QUESTIONS WITH
VERIFIED ANSWERS GRADED A+ TESTED
AND APPROVED!!!
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."
(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
(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. (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
(8.5) Suppose X1, ..., Xn are i.i.d. Bern(p). Find the MLE for p. -- ANSWER--
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(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)? -- ANSWER--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. -- ANSWER--True
(8.6) TRUE or FALSE? Sometimes it might be difficult to obtain an MLE in
closed form. -- ANSWER--True. (There is a gamma example.)
(8.7) Suppose that the MLE for a parameter θ is θhat=4. Find the MLE for √θ. --
ANSWER--
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]? -- ANSWER--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
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hypothesis of lognormality? -- ANSWER--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? --
ANSWER-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? --
ANSWER--Reject. The test statistic 9.12 is not less than 5.99.
(8.12) Consider the PRN's U1 = 0.1 , U2 = 0.9 , and U3 = 0.2. Use
Kolmogorov-Smirnov with α = 0.05 to test to see if these numbers are indeed
uniform. Do we ACCEPT (i.e., fail to reject) or REJECT uniformity? --
ANSWER--Accept. From table, D(α=0.05, 3) = 0.70760. Create ordered sample
set: 0.1, 0.2, 0.9. Since the max value of D test is 0.467, then we fail to reject
because it is smaller.
(9.1) TRUE or FALSE? Simulation output (e.g., consecutive customer waiting
times) is almost never i.i.d. normal - and that's a big fat problem! -- ANSWER--
True
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(9.1) We often distinguish between two general types of simulations with regard
to output analysis. What are they called? -- ANSWER--Finite-horizon and
steady-state
We are interested in modeling the arrival and service process at the local
McBurger Queen burger joint. Customers come in every once in a while, stand
in line, eventually get served, and off they go. Generally speaking, what kind of
model are we talking about here? (More than one answer below may be right.) -
- ANSWER--Discrete (because events such as arrivals and service completions
only happen once in a while, as opposed to continuously) and
Stochastic (because customer arrival times, service times, shift changes, etc., are
all random)
Which of the following can be regarded as advantages of simulation? (More
than one answer below may be right.) -- ANSWER--Simulation enables you to
study models too complicated for analytical or numerical treatment.
Simulations can serve as very pretty demos.
Simulation can be used to study detailed relations that might be lost in an
analytical or numerical treatment.
Who is William Gosset? -- ANSWER--He invented the t distribution that is
used ubiquitously in statistics.
SIMULATION MODULE 1 ISYE 6644
PRACTICE EXAM QUESTIONS WITH
VERIFIED ANSWERS GRADED A+ TESTED
AND APPROVED!!!
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."
(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
(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. (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
(8.5) Suppose X1, ..., Xn are i.i.d. Bern(p). Find the MLE for p. -- ANSWER--
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(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)? -- ANSWER--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. -- ANSWER--True
(8.6) TRUE or FALSE? Sometimes it might be difficult to obtain an MLE in
closed form. -- ANSWER--True. (There is a gamma example.)
(8.7) Suppose that the MLE for a parameter θ is θhat=4. Find the MLE for √θ. --
ANSWER--
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]? -- ANSWER--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
,Page 3 of 54
hypothesis of lognormality? -- ANSWER--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? --
ANSWER-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? --
ANSWER--Reject. The test statistic 9.12 is not less than 5.99.
(8.12) Consider the PRN's U1 = 0.1 , U2 = 0.9 , and U3 = 0.2. Use
Kolmogorov-Smirnov with α = 0.05 to test to see if these numbers are indeed
uniform. Do we ACCEPT (i.e., fail to reject) or REJECT uniformity? --
ANSWER--Accept. From table, D(α=0.05, 3) = 0.70760. Create ordered sample
set: 0.1, 0.2, 0.9. Since the max value of D test is 0.467, then we fail to reject
because it is smaller.
(9.1) TRUE or FALSE? Simulation output (e.g., consecutive customer waiting
times) is almost never i.i.d. normal - and that's a big fat problem! -- ANSWER--
True
, Page 4 of 54
(9.1) We often distinguish between two general types of simulations with regard
to output analysis. What are they called? -- ANSWER--Finite-horizon and
steady-state
We are interested in modeling the arrival and service process at the local
McBurger Queen burger joint. Customers come in every once in a while, stand
in line, eventually get served, and off they go. Generally speaking, what kind of
model are we talking about here? (More than one answer below may be right.) -
- ANSWER--Discrete (because events such as arrivals and service completions
only happen once in a while, as opposed to continuously) and
Stochastic (because customer arrival times, service times, shift changes, etc., are
all random)
Which of the following can be regarded as advantages of simulation? (More
than one answer below may be right.) -- ANSWER--Simulation enables you to
study models too complicated for analytical or numerical treatment.
Simulations can serve as very pretty demos.
Simulation can be used to study detailed relations that might be lost in an
analytical or numerical treatment.
Who is William Gosset? -- ANSWER--He invented the t distribution that is
used ubiquitously in statistics.
