ISYE 6644 REVIEW ALL SET QUESTIONS AND
ANSWERS SET A+
✔✔Brownian motion is also known as ... - ✔✔Wiener process
✔✔Let W ( t ) denote a Brownian motion process at time t. What is C o v ( W ( 2 ) , W (
3 ) )? - ✔✔min(s, t)
2
✔✔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. - ✔✔TRUE.
That's the point of this module!
✔✔If 𝑋 is a continuous random variable with c.d.f. 𝐹(𝑥), what's the distribution of 𝐹(𝑋)? -
✔✔Unif(0,1)
✔✔If 𝑈 is a Unif(0,1) random variable, what's the distribution of −1/𝜆𝑙𝑛(1−𝑈)? - ✔✔Exp(
𝜆)
✔✔f 𝑈 is a Unif(0,1) random variable, what's the distribution of 1/3[−ln(U)]^1/2? -
✔✔Weibull, with parameters λ=3 and β=2
✔✔TRUE or FALSE? You can find the inverse c.d.f. Φ^−1(. ) of the standard normal
distribution in closed form. - ✔✔FALSE. You need to use an approximation.
✔✔If 𝑈 is Unif(0,1), what is ⌈6𝑈⌉? (Note that ⌈.⌉ is the round-up function.) - ✔✔A 6-side
die toss
✔✔If 𝑈 is Unif(0,1), what is ⌈𝑙𝑛(𝑈)/𝑙𝑛(5/6)⌉? (Note that ⌈.⌉ is the round-up function.) -
✔✔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. That's the point of this lesson!
✔✔TRUE or FALSE? The convolution method involves sums of random variables. -
✔✔TRUE
✔✔Support that U1 and U2 are PRNs. What's the distribution of U1 + U2? -
✔✔Triangular (0,1,2)
✔✔Suppose that I want to generate a simple Unif(2/3, 1) via A-R. Support I generate a
PRN U1=0.16. Do I accept U1 as my Unif(2/3, 1)? - ✔✔NO. In this example, we only
accept if 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. Super false,
in fact!
✔✔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
✔✔True or False? The A-R algorithm for X~Pois(λ) tells us to generate PRNs until e^-λ
.... Ui for the first time, and then set X=n. - ✔✔TRUE. Yup - that's how you do it (though
sometimes it's a bit tedious if λ is large).
✔✔Unif(0,1) PRNs can be used to generate which of the following random entities? -
✔✔All of the above --- and just about anything else!
✔✔If X is an Exp(λ) random variable with c.d.f. F ( x ) = 1 − e − λ x, what's the
distribution of the random variable 1 − e − λ X? - ✔✔Unif(0,1)
✔✔If U is a Unif(0,1) random variable, what's the distribution of − 1 λ ln ( U )? -
✔✔Exp(λ)
✔✔Suppose that U 1 , U 2 , ... , U 5000 are i.i.d. Unif(0,1) random variables. Using
Excel (or your favorite programming language), simulate X i = − ln ( U i ) for i = 1 , 2 ,
... , 5000. Draw a histogram of the 5000 numbers. What p.d.f. does the histogram look
like? - ✔✔Exponential
✔✔Suppose the c.d.f. of X is F ( x ) = x^, 0 ≤ x ≤ 2. Develop a generator for X and
demonstrate with U = 0.54. - ✔✔X = 2 U^{1/3} = 1.629
ANSWERS SET A+
✔✔Brownian motion is also known as ... - ✔✔Wiener process
✔✔Let W ( t ) denote a Brownian motion process at time t. What is C o v ( W ( 2 ) , W (
3 ) )? - ✔✔min(s, t)
2
✔✔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. - ✔✔TRUE.
That's the point of this module!
✔✔If 𝑋 is a continuous random variable with c.d.f. 𝐹(𝑥), what's the distribution of 𝐹(𝑋)? -
✔✔Unif(0,1)
✔✔If 𝑈 is a Unif(0,1) random variable, what's the distribution of −1/𝜆𝑙𝑛(1−𝑈)? - ✔✔Exp(
𝜆)
✔✔f 𝑈 is a Unif(0,1) random variable, what's the distribution of 1/3[−ln(U)]^1/2? -
✔✔Weibull, with parameters λ=3 and β=2
✔✔TRUE or FALSE? You can find the inverse c.d.f. Φ^−1(. ) of the standard normal
distribution in closed form. - ✔✔FALSE. You need to use an approximation.
✔✔If 𝑈 is Unif(0,1), what is ⌈6𝑈⌉? (Note that ⌈.⌉ is the round-up function.) - ✔✔A 6-side
die toss
✔✔If 𝑈 is Unif(0,1), what is ⌈𝑙𝑛(𝑈)/𝑙𝑛(5/6)⌉? (Note that ⌈.⌉ is the round-up function.) -
✔✔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. That's the point of this lesson!
✔✔TRUE or FALSE? The convolution method involves sums of random variables. -
✔✔TRUE
✔✔Support that U1 and U2 are PRNs. What's the distribution of U1 + U2? -
✔✔Triangular (0,1,2)
✔✔Suppose that I want to generate a simple Unif(2/3, 1) via A-R. Support I generate a
PRN U1=0.16. Do I accept U1 as my Unif(2/3, 1)? - ✔✔NO. In this example, we only
accept if 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. Super false,
in fact!
✔✔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
✔✔True or False? The A-R algorithm for X~Pois(λ) tells us to generate PRNs until e^-λ
.... Ui for the first time, and then set X=n. - ✔✔TRUE. Yup - that's how you do it (though
sometimes it's a bit tedious if λ is large).
✔✔Unif(0,1) PRNs can be used to generate which of the following random entities? -
✔✔All of the above --- and just about anything else!
✔✔If X is an Exp(λ) random variable with c.d.f. F ( x ) = 1 − e − λ x, what's the
distribution of the random variable 1 − e − λ X? - ✔✔Unif(0,1)
✔✔If U is a Unif(0,1) random variable, what's the distribution of − 1 λ ln ( U )? -
✔✔Exp(λ)
✔✔Suppose that U 1 , U 2 , ... , U 5000 are i.i.d. Unif(0,1) random variables. Using
Excel (or your favorite programming language), simulate X i = − ln ( U i ) for i = 1 , 2 ,
... , 5000. Draw a histogram of the 5000 numbers. What p.d.f. does the histogram look
like? - ✔✔Exponential
✔✔Suppose the c.d.f. of X is F ( x ) = x^, 0 ≤ x ≤ 2. Develop a generator for X and
demonstrate with U = 0.54. - ✔✔X = 2 U^{1/3} = 1.629
