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ISYE6644 MIDTERM 2 LATEST 2025 ACTUAL EXAM WITH COMPLETE
QUESTIONS AND CORRECT DETAILED ANSWERS (100% VERIFIED
ANSWERS) |ALREADY GRADED A+| ||PROFESSOR VERIFIED||
Suppose that X is a continuous RV with p.d.f. f(x) = 30x^4(1-x) for
0<x<1. Why is acceptance-rejection a good method to use to
generate X? - ANSWER-Because the c.d.f. of X is very hard to
invert.
Unif(0,1) PRNs can be used to generate which of the following
random entities? - ANSWER-Exp(lambda) random variates,
Nor(0,1) random variates, Triangular random variates, Bern(p)
random variates, Nonhomogeneous Poisson processes, and just
about anything else.
If X is an Exp(lambda) random variable with c.d.f. F(x) = 1-e^(-
lambdax), what's the distribution of the random variable 1-e^(-
lambdaX)? - ANSWER-Unif(0,1). Inverse transform theroem.
If U is a Unif(0,1) random variable, what's the distribution of -
1/lambda(ln(U))? - ANSWER-Exp(lambda)
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If X is a Nor(0,1) random variate, and Φ ( x ) is the Nor(0,1) c.d.f.,
what is the distribution of Φ ( X )? - ANSWER-Uniform. By the
inverse transform theorem, Φ ( X ) ~ Unif(0,1).
How would you simulate the sum of two 6-sided dice tosses? -
ANSWER-[6U1] + [6U2]
If U is Unif(0,1), how can we simulate a Geom(0.6) random
variate? - ANSWER-[ln(U)/ln(0.4)] or [ln(1-U)/ln(0.4)]
Suppose that U and V are PRNs. Let X=U+V. Simulate this 5000
times, and draw a histogram of the 5000 numbers. What p.d.f.
does the histogram look like? - ANSWER-Triangular.
In general, the majorizing function t(x) is itself a p.d.f. f(x)? -
ANSWER-False
Suppose that X is a continuous RV with p.d.f. f(x) = 30x^4(1-x) for
0<x<1. What's a good method that you can use to generate a
realization of X? - ANSWER-Acceptance-Rejection
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Consider four observations from some unknown distribution,
X1=1.5, X2 = -3.7, X3=2.7, and X4= 0.6. What is the fourth order
statistic, denoted X(4)? - ANSWER-2.7. X(4) means the largest of
the sample of 4 observations.
Which of the following are true? - ANSWER-The closer the
majorizing function t(x) is to the true p.d.f. f(x), the more efficient
the A-R algorithm is; h(y) = t(y)/integral of t(x) is itself a p.d.f.,
random variates from h(y) should be "easy" to generate
Suppose that an airline can experience three types of departure
delays - A, B, and C, which occur with probabilities 0.4, 0.1, and
0.5, respectively. Delays A, B, and C are exponential, normal, and
Weibull, respectively. YES or NO? Would the composition method
be a good choice to generate a delay time for this scenario? -
ANSWER-Yes. Composition would be much more efficient than
figuring out and delaying with the crazy c.d.f. of the overall delay.
Let H and W denote a person's height and weight. Which of the
following best describes the joint distribution of (H,W)? -
ISYE6644 MIDTERM 2 LATEST 2025 ACTUAL EXAM WITH COMPLETE
QUESTIONS AND CORRECT DETAILED ANSWERS (100% VERIFIED
ANSWERS) |ALREADY GRADED A+| ||PROFESSOR VERIFIED||
Suppose that X is a continuous RV with p.d.f. f(x) = 30x^4(1-x) for
0<x<1. Why is acceptance-rejection a good method to use to
generate X? - ANSWER-Because the c.d.f. of X is very hard to
invert.
Unif(0,1) PRNs can be used to generate which of the following
random entities? - ANSWER-Exp(lambda) random variates,
Nor(0,1) random variates, Triangular random variates, Bern(p)
random variates, Nonhomogeneous Poisson processes, and just
about anything else.
If X is an Exp(lambda) random variable with c.d.f. F(x) = 1-e^(-
lambdax), what's the distribution of the random variable 1-e^(-
lambdaX)? - ANSWER-Unif(0,1). Inverse transform theroem.
If U is a Unif(0,1) random variable, what's the distribution of -
1/lambda(ln(U))? - ANSWER-Exp(lambda)
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If X is a Nor(0,1) random variate, and Φ ( x ) is the Nor(0,1) c.d.f.,
what is the distribution of Φ ( X )? - ANSWER-Uniform. By the
inverse transform theorem, Φ ( X ) ~ Unif(0,1).
How would you simulate the sum of two 6-sided dice tosses? -
ANSWER-[6U1] + [6U2]
If U is Unif(0,1), how can we simulate a Geom(0.6) random
variate? - ANSWER-[ln(U)/ln(0.4)] or [ln(1-U)/ln(0.4)]
Suppose that U and V are PRNs. Let X=U+V. Simulate this 5000
times, and draw a histogram of the 5000 numbers. What p.d.f.
does the histogram look like? - ANSWER-Triangular.
In general, the majorizing function t(x) is itself a p.d.f. f(x)? -
ANSWER-False
Suppose that X is a continuous RV with p.d.f. f(x) = 30x^4(1-x) for
0<x<1. What's a good method that you can use to generate a
realization of X? - ANSWER-Acceptance-Rejection
, 3|Page
Consider four observations from some unknown distribution,
X1=1.5, X2 = -3.7, X3=2.7, and X4= 0.6. What is the fourth order
statistic, denoted X(4)? - ANSWER-2.7. X(4) means the largest of
the sample of 4 observations.
Which of the following are true? - ANSWER-The closer the
majorizing function t(x) is to the true p.d.f. f(x), the more efficient
the A-R algorithm is; h(y) = t(y)/integral of t(x) is itself a p.d.f.,
random variates from h(y) should be "easy" to generate
Suppose that an airline can experience three types of departure
delays - A, B, and C, which occur with probabilities 0.4, 0.1, and
0.5, respectively. Delays A, B, and C are exponential, normal, and
Weibull, respectively. YES or NO? Would the composition method
be a good choice to generate a delay time for this scenario? -
ANSWER-Yes. Composition would be much more efficient than
figuring out and delaying with the crazy c.d.f. of the overall delay.
Let H and W denote a person's height and weight. Which of the
following best describes the joint distribution of (H,W)? -
