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ISyE 6644 — Summer 2009 — Test #1 Solutions
This test is open book, open notes. You have 90 minutes. Good luck!
- 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
- 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
