ISYE 6644 STUDYS ACTUAL EXAM QUESTIONS AND
ANSWERS SURE A+
✔✔If U is a Unif(0,1) random variable, what's the distribution of 1/3[-ln(U)]^1/2? -
✔✔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. - ✔✔False. You need to use an approximation.
✔✔If U is Unif(0,1), what is [6U]? - ✔✔A 6-side die toss
✔✔If U is Unif(0,1), what is [ln(U)/ln(5/6)]? - ✔✔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. -
✔✔True
✔✔TRUE or FALSE? The convolution method involves sums of random variables. -
✔✔True
✔✔Suppose that U1 and U2 are PRNs. Whats the distribution of U1 + U2? -
✔✔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)? -
✔✔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. - ✔✔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? - ✔✔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? -
✔✔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)? - ✔✔Unif(0,1). Inverse transform
theroem.
✔✔If U is a Unif(0,1) random variable, what's the distribution of -1/lambda(ln(U))? -
✔✔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 )? - ✔✔Uniform. By the inverse transform theorem, Φ ( X ) ~
Unif(0,1).
✔✔How would you simulate the sum of two 6-sided dice tosses? - ✔✔[6U1] + [6U2]
✔✔If U is Unif(0,1), how can we simulate a Geom(0.6) random variate? -
✔✔[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? -
✔✔Triangular.
✔✔In general, the majorizing function t(x) is itself a p.d.f. f(x)? - ✔✔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? - ✔✔Acceptance-
Rejection
✔✔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)? - ✔✔2.7. X(4)
means the largest of the sample of 4 observations.
✔✔Which of the following are true? - ✔✔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
ANSWERS SURE A+
✔✔If U is a Unif(0,1) random variable, what's the distribution of 1/3[-ln(U)]^1/2? -
✔✔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. - ✔✔False. You need to use an approximation.
✔✔If U is Unif(0,1), what is [6U]? - ✔✔A 6-side die toss
✔✔If U is Unif(0,1), what is [ln(U)/ln(5/6)]? - ✔✔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. -
✔✔True
✔✔TRUE or FALSE? The convolution method involves sums of random variables. -
✔✔True
✔✔Suppose that U1 and U2 are PRNs. Whats the distribution of U1 + U2? -
✔✔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)? -
✔✔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. - ✔✔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? - ✔✔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? -
✔✔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)? - ✔✔Unif(0,1). Inverse transform
theroem.
✔✔If U is a Unif(0,1) random variable, what's the distribution of -1/lambda(ln(U))? -
✔✔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 )? - ✔✔Uniform. By the inverse transform theorem, Φ ( X ) ~
Unif(0,1).
✔✔How would you simulate the sum of two 6-sided dice tosses? - ✔✔[6U1] + [6U2]
✔✔If U is Unif(0,1), how can we simulate a Geom(0.6) random variate? -
✔✔[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? -
✔✔Triangular.
✔✔In general, the majorizing function t(x) is itself a p.d.f. f(x)? - ✔✔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? - ✔✔Acceptance-
Rejection
✔✔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)? - ✔✔2.7. X(4)
means the largest of the sample of 4 observations.
✔✔Which of the following are true? - ✔✔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
