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Solutions to Exam 1 - Simulation | ISYE 6644, Exams of Systems Engineering

Material Type: Exam; Class: Simulation; Subject: Industrial & Systems Engr; University: Georgia Institute of Technology-Main Campus; Term: Summer 2009;

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ISyE 6644 Summer 2009 Test #1 Solutions
This test is open book, open notes. You have 90 minutes. Good luck!
1. Short-answer probability questions.
(a) Suppose Xhas p.d.f. f(x) = 3x2, 0 < x < 1. Find E[3X+ 2].
Solution: E[X] = R1
0x3x2dx = 3/4. Thus, E[3X+ 2] = 1/4. 2
(b) Suppose Xhas p.d.f. f(x) = 3x2, 0 < x < 1. Find E[1/X].
Solution: E[1/X] = R1
0(1/x)3x2dx =R1
03x dx = 3/2. 2
(c) Suppose Xis a random variable such that E[Xn] = n!/2n,n= 1,2, . . .. Find
Var(X2).
Solution: Var[X2] = E[X4](E[X2])2= 4!/24(2!/22)2= 5/4. 2
(d) Suppose Xhas m.g.f. MX(t) = et2. Find E[X].
Solution: E[X] = d
dt MX(t)|t=0 = 2tet2|t=0 = 0. 2
(e) TRUE or FALSE? Suppose Xhas m.g.f. MX(t). If Y=aX +b, then the
m.g.f. of Yis etbMX(at).
Solution: True. MY(t) = E[etY ] = E[eatX+tb] = etbMX(at). 2
(f) TRUE or FALSE? The customer arrival process at a restaurant is well-
represented by a Poisson process with a constant rate function.
Solution: False. Arrival rate changes throughout the day. 2
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ISyE 6644 — Summer 2009 — Test #1 Solutions

This test is open book, open notes. You have 90 minutes. Good luck!

  1. Short-answer probability questions.

(a) Suppose X has p.d.f. f (x) = 3x^2 , 0 < x < 1. Find E[− 3 X + 2].

Solution: E[X] =

∫ (^1) 0 x^3 x (^2) dx = 3/4. Thus, E[− 3 X + 2] = − 1 /4. 2

(b) Suppose X has p.d.f. f (x) = 3x^2 , 0 < x < 1. Find E[1/X].

Solution: E[1/X] =

∫ (^1) 0 (1/x)3x

(^2) dx = ∫^1 0 3 x dx^ = 3/2.^2

(c) Suppose X is a random variable such that E[Xn] = n!/ 2 n, n = 1, 2 ,.. .. Find Var(X^2 ).

Solution: Var[X^2 ] = E[X^4 ] − (E[X^2 ])^2 = 4!/ 24 − (2!/ 22 )^2 = 5/4. 2

(d) Suppose X has m.g.f. MX (t) = et 2

. Find E[X].

Solution: E[X] = (^) dtd MX (t)|t=0 = 2tet

2 |t=0 = 0. 2

(e) TRUE or FALSE? Suppose X has m.g.f. MX (t). If Y = aX + b, then the m.g.f. of Y is etbMX (at).

Solution: True. MY (t) = E[etY^ ] = E[eatX+tb] = etbMX (at). 2

(f) TRUE or FALSE? The customer arrival process at a restaurant is well- represented by a Poisson process with a constant rate function.

Solution: False. Arrival rate changes throughout the day. 2

(g) Suppose that machine breakdowns occur according to a Poisson process at the rate of 1/week. What’s the probability that there will be no breakdowns during the next two weeks?

Solution: Let X be the number of breakdowns in 2 seeks. Then X ∼ Pois(λ = 2 / 2 weeks). P(X = 0) = e

−λλ 0 0! =^ e

(h) TRUE or FALSE? The times between arrivals of a Poisson process are i.i.d. exponential.

Solution: True. 2

(i) Suppose that Y has p.d.f. f (y) = 2y, 0 ≤ y ≤ 1. Find the p.d.f. of Z = Y 2.

Solution: The c.d.f. of Z is G(z) = P(Z ≤ z) = P(Y ≤

z) =

∫ √z 0 2 y dy^ =^ z. Thus, the p.d.f. of Z is g(z) = 1 for 0 ≤ z ≤ 1. (In other words, Z ∼ U (0, 1).) 2

(j) Suppose X is an Exp(λ = 3) random variable. What’s the distribution of the nasty-looking random variable Y = 1 − e−^3 X^?

Solution: By the inverse transform theorem, Y ∼ U (0, 1) 2

(k) Suppose the joint p.d.f. of X and Y is f (x, y) = cx, 0 < x < y < 1, for some appropriate value of c. Find E[X].

Solution: Find c first.

∫ (^1)

0

∫ (^) y

0

cx dx dy =

c 2

∫ (^1)

0

y^2 dy = c/ 6 ,

so that c = 6. Then the p.d.f. of X is

fX (x) =

R

f (x, y) dy =

∫ (^1)

x

6 x dy = 6x(1 − x), 0 < x < 1.

Thus, E[X] =

∫ (^1) 0 x^6 x(1^ −^ x)^ dx^ = 1/2.^2

Solution: − 3 `n(Ui), i = 1, 2 , are i.i.d. Exp(1/3). Thus, the sum of the two is Erlang 2 (1/3). 2

(s) Suppose that I toss 1000 darts randomly onto a unit square, on which I’ve inscribed a circle with radius 1/2. Suppose that 755 of those darts land inside the circle. Using this information, give me an estimate of π.

Solution: From class, ˆπ = 4ˆp = 4(755/1000) = 3.02. 2

(t) What does “LCG” mean?

Solution: Linear Congruential Generator. 2

(u) Suppose that X 0 = 0 is the integer seed for the pseudo-random number gen- erator Xi = (5Xi− 1 + 3)mod(8). Find X 32.

Solution: X 0 = 0, X 1 = 3, X 2 = 2, X 3 = 5, X 4 = 4, X 5 = 7, X 6 = 6, X 7 = 1, X 8 = 0, so that the cycle length is 8. Thus, X 32 = 0. 2

(v) TRUE or FALSE? Although Arena uses a Process-Interaction modeling phi- losophy, deep down inside its code, it carries out Event-Scheduling.

Solution: True. 2

(w) In Arena, what is the mean of the expression EXPO(3)?

Solution: 3. 2

(x) TRUE or FALSE? In an Arena PROCESS block, it is possible to do a SEIZE- DELAY without a RELEASE. (Gasp!)

Solution: True. 2

(y) Who is the best teacher ever? (No partial credit will be given for this ques- tion.)

Solution: Christos Alexopoulos. 2

  1. Joey works at an ice cream shop. Six customer interarrival times are as follows (in minutes):

Customers are served in LIFO fashion. The 6 customers order the following num- bers of ice cream products, respectively:

Suppose it takes Joey 3 minutes to prepare each ice cream product. Further suppose that he charges $3/ice cream. Sadly, the customers are unruly and each customer causes $1 in damage for every minute the customer has to wait in line.

(a) When does the first customer leave?

Solution: cust arrl time start serv serve time depart wait 1 1 1 18 19 0 2 3 46 6 52 43 3 8 37 9 46 29 4 10 34 3 37 24 5 12 22 12 34 10 6 18 19 3 22 1 Thus, the first customer leaves at time 19. 2

(b) Which customer is the second guy to be served?

Solution: Customer 6. 2

(c) What is the average number of customers in the system during the first 10 minutes?

Solution: Plot the number of customers in the system, L(t), and compute 1 10

∫ (^10) 0 L(t)^ dt^ = 1.8.^2