ISYE 6644 TEST PAPER EXAM QUESTIONS AND
ANSWERS SET A+
✔✔How would you simulate the sum of two 6-sided dice tosses? (Note that ⌈ ⋅ ⌉ is the
round-up function; and all of the U's denote PRNs.) - ✔✔⌈ 6 U 1 ⌉ + ⌈ 6 U 2 ⌉
✔✔If U is Unif(0,1), how can we simulate a Geom(0.6) random variate? - ✔✔⌈ ln ( U )
/ ln ( 0.4 ) ⌉
⌈ ln ( 1 − U ) / ln ( 0.4 ) ⌉
✔✔Consider four observations from some unknown distribution, X 1 = 1.5, X 2 = − 3.7,
X 3 = 2.7, and X 4 = 0.6. What is the fourth order statistic, which we denoted by X ( 4 )
in class? - ✔✔2.7
merely means the largest of the sample of 4 observations.
✔✔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
By the lesson notes, we know that the 5000 Xi's are all Triangular(0,1,2). Since we have
a histogram of 5000 of these, it really ought to look like a triangular p.d.f. Thus, the
answer is (c).
✔✔Suppose that U 1 , U 2 , ... , U 24 are i.i.d. PRNs. What is the approximate
distribution of X = 5 + 3 ∑ i = 1 24 U i? - ✔✔Nor(41,18)
✔✔In general, the majorizing function t ( x ) is itself a p.d.f. f ( x ). - ✔✔False
the majorizing function generally integrates to a number greater than 1, and so it cannot
be a legitimate p.d.f.
, ✔✔Suppose that X is a continuous RV with p.d.f. f ( x ) = 30 x 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 the constant c = ∫ R t ( x ) d x = 5. On average, how many iterations (trials)
will the A-R algorithm require? - ✔✔5
✔✔ What is a possible goal of an indifference-zone normal means selection technique?
- ✔✔Find the normal population having the largest mean, especially if the largest mean
is ≫ the second-largest.
✔✔TRUE or FALSE? The Bechhofer procedure for selecting the normal population with
the largest mean specifies the appropriate number of observations to take from each
competing population, and simply selects the competitor having the largest sample
mean. - ✔✔True
✔✔TRUE or FALSE? Sometimes a single-stage procedure like Bechhofer's is
inefficient. In fact, it's possible to use certain sequential procedures that take
observations one-at-a-time (instead of all at once in a single stage) to make good
selection decisions using fewer observations. - ✔✔True
✔✔For which scenarios(s) below might it be appropriate to use a Bernoulli selection
procedure?
a) Find the inventory policy having the largest profit.
b) Find the drug giving the best chance of a cure.
c) Find the maintenance policy having the lowest failure probability.
d) Find the scheduling rule that that has the best chance of making an on-time delivery.
- ✔✔All three of (b), (c), and (d).
✔✔Suppose that a Bernoulli selection procedure tells you to take 100 observations from
each of two populations, A and B. It turns out that A gets 85 successes and B gets 46
successes. What do you think? - ✔✔1) A almost certainly has a higher success
probability than B.
2) We could've probably stopped sampling a bit earlier (i.e., with fewer than 100
observations) because A was so far ahead of B.
✔✔For which scenarios(s) below might it be appropriate to use a multinomial selection
procedure? - ✔✔Find the most-popular political candidate.
✔✔Suppose that we want to know which of Coke, Pepsi and Dr.pepper is the most
popular. We would like to make the correct selection with probability of at least P*=0.90
in the event that the ration of the highest-to-second-highest preference probabilities
happens to be at least 0*=1.4. How many people does the single-stage procedure
Mbem require us to interview? - ✔✔126
ANSWERS SET A+
✔✔How would you simulate the sum of two 6-sided dice tosses? (Note that ⌈ ⋅ ⌉ is the
round-up function; and all of the U's denote PRNs.) - ✔✔⌈ 6 U 1 ⌉ + ⌈ 6 U 2 ⌉
✔✔If U is Unif(0,1), how can we simulate a Geom(0.6) random variate? - ✔✔⌈ ln ( U )
/ ln ( 0.4 ) ⌉
⌈ ln ( 1 − U ) / ln ( 0.4 ) ⌉
✔✔Consider four observations from some unknown distribution, X 1 = 1.5, X 2 = − 3.7,
X 3 = 2.7, and X 4 = 0.6. What is the fourth order statistic, which we denoted by X ( 4 )
in class? - ✔✔2.7
merely means the largest of the sample of 4 observations.
✔✔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
By the lesson notes, we know that the 5000 Xi's are all Triangular(0,1,2). Since we have
a histogram of 5000 of these, it really ought to look like a triangular p.d.f. Thus, the
answer is (c).
✔✔Suppose that U 1 , U 2 , ... , U 24 are i.i.d. PRNs. What is the approximate
distribution of X = 5 + 3 ∑ i = 1 24 U i? - ✔✔Nor(41,18)
✔✔In general, the majorizing function t ( x ) is itself a p.d.f. f ( x ). - ✔✔False
the majorizing function generally integrates to a number greater than 1, and so it cannot
be a legitimate p.d.f.
, ✔✔Suppose that X is a continuous RV with p.d.f. f ( x ) = 30 x 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 the constant c = ∫ R t ( x ) d x = 5. On average, how many iterations (trials)
will the A-R algorithm require? - ✔✔5
✔✔ What is a possible goal of an indifference-zone normal means selection technique?
- ✔✔Find the normal population having the largest mean, especially if the largest mean
is ≫ the second-largest.
✔✔TRUE or FALSE? The Bechhofer procedure for selecting the normal population with
the largest mean specifies the appropriate number of observations to take from each
competing population, and simply selects the competitor having the largest sample
mean. - ✔✔True
✔✔TRUE or FALSE? Sometimes a single-stage procedure like Bechhofer's is
inefficient. In fact, it's possible to use certain sequential procedures that take
observations one-at-a-time (instead of all at once in a single stage) to make good
selection decisions using fewer observations. - ✔✔True
✔✔For which scenarios(s) below might it be appropriate to use a Bernoulli selection
procedure?
a) Find the inventory policy having the largest profit.
b) Find the drug giving the best chance of a cure.
c) Find the maintenance policy having the lowest failure probability.
d) Find the scheduling rule that that has the best chance of making an on-time delivery.
- ✔✔All three of (b), (c), and (d).
✔✔Suppose that a Bernoulli selection procedure tells you to take 100 observations from
each of two populations, A and B. It turns out that A gets 85 successes and B gets 46
successes. What do you think? - ✔✔1) A almost certainly has a higher success
probability than B.
2) We could've probably stopped sampling a bit earlier (i.e., with fewer than 100
observations) because A was so far ahead of B.
✔✔For which scenarios(s) below might it be appropriate to use a multinomial selection
procedure? - ✔✔Find the most-popular political candidate.
✔✔Suppose that we want to know which of Coke, Pepsi and Dr.pepper is the most
popular. We would like to make the correct selection with probability of at least P*=0.90
in the event that the ration of the highest-to-second-highest preference probabilities
happens to be at least 0*=1.4. How many people does the single-stage procedure
Mbem require us to interview? - ✔✔126
