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ISyE 6644 — Spring 2018 — Practice Test
(revised 6/18/21)
Hi Class!
This test was originally given in my “live” 6644 class during the Fall 2017 semester. The
students had 75 minutes, and were allowed to bring a cheat sheet (both sides). On our
Test 1, I’ll be more generous with time and cheat sheets.
Also, all of our test questions will be multiple choice, in spite of what you see below.
Dave
- Toss two dice and observe their sum. What is the expected number of tosses until you observe a sum of 11?
Solution: X ∼ Geom(1/18), so E[X] = 18.
- Consider a Poisson process with rate λ = 2. What is the distribution of the time between the 5rd and 6th arrivals?
Solution: Exp(2).
- YES or NO? If X and Y are independent exponential random variables, are they necessarily uncorrelated?
Solution: Yes.
- TRUE or FALSE? A Poisson process has stationary and independent increments.
Solution: True.
- Suppose X has p.d.f. f (x) = 4 x^3 , 0 < x < 1. Find E[ 2 X −^ 3].
Solution:
E
X
0
x
4 x 3 dx =
0
4 x 2 dx = 4 / 3.
This implies that
E
X
= 2 E
X
- Suppose that X has p.d.f. f (x) = 3 x^2 , 0 ≤ x ≤ 1. What’s the p.d.f. of the random
variable eX^?
Solution: Let Y = e X
. The c.d.f. of Y is
G(y) = P(Y ≤ y) = P(e X ≤ y) = P(X ≤ n(y)) =
(^) n(y)
0
3 x 2 dx = [n(y)] 3 .
This implies that the p.d.f. of Y is
g(y) =
d
dy
G(y) =
3[n(y)] 2
y
, 1 ≤ y ≤ e.
- Suppose that X and Y have joint p.d.f. f (x, y) = 8 xy for 0 < y < x < 1.
(a) Find E[X].
(b) Find Cov(X, Y ).
Solution: We have
fX (x) =
R
f (x, y) dy =
(^) x
0
8 xy dy = 4 x 3 , 0 < x < 1 ,
so that
E[X] =
R
xfX (x) dx =
0
4 x 4 dx = 4 / 5.
Similarly,
fY (y) =
R
f (x, y) dx =
y
8 xy dx = 4(y − y 3 ), 0 < y < 1 ,
- BONUS: What beloved American actress had her birthday on Valentine’s Day?
Solution: Florence Henderson. (Sadly, Florence passed away last November. )
- Suppose that the Atlanta Hawks play i.i.d. games, each of which has win proba-
bility 0.6. Let X be the number of games until the Hawks achieve their first win. Find the smallest x such that P(X ≤ x) ≥ 0 .9.
Solution: The number of games until the first win is X ∼ Geom(0.6), so that P(X = x) = q x−^1 p = (0.4)x−^1 (0.6). It can also be shown that the c.d.f. is
F (x) = P(X ≤ x) = 1 − q x = 1 − (0.4) x
(though you don’t need to know this if you do a trial-and-error argument to find the smallest x). Now, F (x) = 1 − (0.4)x^ ≥ 0. 9 iff 0. 1 ≥ (0.4)x^ , which is achieved by x = 3.
- Suppose X 1 ,... , Xn are i.i.d. from some distribution. What tells us that the
sample mean of the Xi’s is approximately normal for large enough n?
Solution: CLT.
- Suppose that X 1 ,... , X 100 are i.i.d. with values 1 and −1, each with probability
0.5. (This is a simple random walk.) Find the approximate probability that the sum
i=1 Xi^ will^ be^ at^ least^ 10.
Solution: Note that E[Xi] = 0 and Var(Xi) = E[X 2 i ]^ −^ (E[Xi])
2 = 1. Then the Central Limit Theorem implies that
i=1 Xi^ ≈^ Nor(0,^ 100),^ and^ so
P(
^100
i=
Xi > 10) ≈ P
Z >
= P(Z > 1) = 0. 1587.
- Consider the linear congruential generator Xi+1 = (3Xi + 1)mod(8).
(a) Using X 0 = 1, calculate the first pseudo-random number U 1.
Solution: We immediately have X 1 = 4, so that U 1 = 0 .5.
(b) Using X 0 = 1, calculate the pseudo-random number U 801.
Solution: If X 0 = 1, then we get X 1 = 4, X 2 = 5, X 3 = 0, and X 4 = 1, so that the thing repeats every 4 tries. Thus, X 801 = X 1 = 4, so that U 1 = 0 .5.
- If U ∼ Unif(0, 1), what’s the distribution of − 3 n(U )?
Solution: Exp(1/3).
- If U 1 and U 2 are i.i.d. Unif(0,1), what’s the distribution of − 3 n(U 1 U 2 )?
Solution: Note that − 3 n(U 1 U 2 ) = − 3 n(U 1 ) − 3 n(U 2 ). Thus, Erlang 2 (1/3) (or gamma).
- If U 1 and U 2 are i.i.d. Unif(0,1), what’s the distribution of 2+
− 2 n(U 1 )cos(2πU 2 )?
Solution: Nor(2,1).
- Suppose that you want to estimate the integral
I =
0
[1 + cos(πx)]dx.
The following numbers are a Unif(0,1) sample:
Use the Monte Carlo method from class to approximate the integral via the estimator I¯ 4.
Suppose it takes Joey 3 minutes to prepare each chocolate product. Further suppose that he charges $2/chocolate. Unfortunately, the customers are unruly and each customer causes $0.50 in damage for every minute the customer has to wait in line.
(a) When does the first customer leave?
(b) What is the average number of customers in the system during the first 20 minutes?
(c) How much money will Joey make or lose with the above 4 customers?
Solution: Consider the following table.
cust intrarrl arrl time serv start serve time depart wait sys time 1 8 8 8 18 26 0 18 2 2 10 38 6 44 28 34 3 5 15 29 9 38 14 23 4 2 17 26 3 29 9 12
(a) The first customer leaves at time 26.
(b) Let Xi denote the amount of time Customer i spends in the system during the time interval [0,20]. In particular, X 1 = 20 − 8 = 12, X 2 = 20 − 10 = 10, X 3 = 5, and X 4 = 3. The average number of customers in the system during the first 20 minutes is
total customer time
20
(c) Joey makes 2(6 + 2 + 3 + 1) − 0 .5(0 + 28 + 14 + 9) = −$1.50.
- Note that I may also ask some Module 4 trivia questions as well as some very
easy Module 5 Arena questions, similar to those you encountered on your lesson assessments and weekly HWs.
