ISYE 6644 Midterm Exam 200 Practice
Questions with Correct Verified Answers / Isye
6644 Midterm Exam Correctly Answered
Practice Questions Latest 2025-2026 (New!)
YES or NO? Does Xi = (Xi-1 + 12)mod(13) have full period? - ANSWER-Yes, it
trivially cycles through 0,1,2,....,11
Which uniform generator was recommended in class, at least as a desert island
generator? - ANSWER-Xi = 16807mod(2^31 - 1)
TRUE or FALSE? Unif(0,1) pseudo-random numbers can be used to generate
pretty much any other random variates, e.g. exponential, normal, and Poisson. -
ANSWER-True
If X is a continuous variable with c.d.f. F(x), what's the distribution of F(X)? -
ANSWER-Unif(0,1). Inverse transform theorem.
If U is a Unif(0,1) random variable, what's the distribution of -1/lambda(ln(1-U))?
- ANSWER-Exp(lambda)
If U is a Unif(0,1) random variable, what's the distribution of 1/3[-ln(U)]^1/2? -
ANSWER-Weibull, with parameters lambda =3 and beta =2
TRUE or FALSE? You can find the inverse c.d.f. of the standard normal
distribution in closed form. - ANSWER-False. You need to use an approximation.
pg. 1
,If U is Unif(0,1), what is [6U]? - ANSWER-A 6-side die toss
If U is Unif(0,1), what is [ln(U)/ln(5/6)]? - ANSWER-Geom(1/6)
TRUE or FALSE? If you can't find a good theoretical distribution to model a
certain random variable, you might want to use the empirical distribution of the
data to do so. - ANSWER-True
TRUE or FALSE? The convolution method involves sums of random variables. -
ANSWER-True
Suppose that U1 and U2 are PRNs. Whats the distribution of U1 + U2? -
ANSWER-Triangular(0,1,2)
YES or NO? As in the notes, suppose that I want to generate a simple Unif(2/3,1)
via A-R. Suppose I generate a PRN U1 = 0.16. Do I accept U1 as my Unif(2/3,1)?
- ANSWER-No. In this example, we only accept U1 >= 2/3; so we reject and try
again until we meet that condition.
TRUE or FALSE? The proof that A-R works is really easy. - ANSWER-False
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.
pg. 2
, 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 theorem.
If U is a Unif(0,1) random variable, what's the distribution of -1/lambda(ln(U))? -
ANSWER-Exp(lambda)
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
pg. 3
Questions with Correct Verified Answers / Isye
6644 Midterm Exam Correctly Answered
Practice Questions Latest 2025-2026 (New!)
YES or NO? Does Xi = (Xi-1 + 12)mod(13) have full period? - ANSWER-Yes, it
trivially cycles through 0,1,2,....,11
Which uniform generator was recommended in class, at least as a desert island
generator? - ANSWER-Xi = 16807mod(2^31 - 1)
TRUE or FALSE? Unif(0,1) pseudo-random numbers can be used to generate
pretty much any other random variates, e.g. exponential, normal, and Poisson. -
ANSWER-True
If X is a continuous variable with c.d.f. F(x), what's the distribution of F(X)? -
ANSWER-Unif(0,1). Inverse transform theorem.
If U is a Unif(0,1) random variable, what's the distribution of -1/lambda(ln(1-U))?
- ANSWER-Exp(lambda)
If U is a Unif(0,1) random variable, what's the distribution of 1/3[-ln(U)]^1/2? -
ANSWER-Weibull, with parameters lambda =3 and beta =2
TRUE or FALSE? You can find the inverse c.d.f. of the standard normal
distribution in closed form. - ANSWER-False. You need to use an approximation.
pg. 1
,If U is Unif(0,1), what is [6U]? - ANSWER-A 6-side die toss
If U is Unif(0,1), what is [ln(U)/ln(5/6)]? - ANSWER-Geom(1/6)
TRUE or FALSE? If you can't find a good theoretical distribution to model a
certain random variable, you might want to use the empirical distribution of the
data to do so. - ANSWER-True
TRUE or FALSE? The convolution method involves sums of random variables. -
ANSWER-True
Suppose that U1 and U2 are PRNs. Whats the distribution of U1 + U2? -
ANSWER-Triangular(0,1,2)
YES or NO? As in the notes, suppose that I want to generate a simple Unif(2/3,1)
via A-R. Suppose I generate a PRN U1 = 0.16. Do I accept U1 as my Unif(2/3,1)?
- ANSWER-No. In this example, we only accept U1 >= 2/3; so we reject and try
again until we meet that condition.
TRUE or FALSE? The proof that A-R works is really easy. - ANSWER-False
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.
pg. 2
, 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 theorem.
If U is a Unif(0,1) random variable, what's the distribution of -1/lambda(ln(U))? -
ANSWER-Exp(lambda)
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
pg. 3
