



Study with the several resources on Docsity
Earn points by helping other students or get them with a premium plan
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
Community
Ask the community for help and clear up your study doubts
Discover the best universities in your country according to Docsity users
Free resources
Download our free guides on studying techniques, anxiety management strategies, and thesis advice from Docsity tutors
Material Type: Exam; Class: Simulation; Subject: Industrial & Systems Engr; University: Georgia Institute of Technology-Main Campus; Term: Summer 2009;
Typology: Exams
Limited-time offer
Uploaded on 08/05/2009
4.8
(3)10 documents
1 / 6
This page cannot be seen from the preview
Don't miss anything!
On special offer
This test is open book, open notes. You have 90 minutes. Good luck!
(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
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