ISyE 3044M Test 1. (a) If X is a continuous random variable with p.d.f. f(x) = 4x 3 for 0 < x < c, find c.

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1 1 NAME ISyE 3044M Test 1 Summer 2002 This test is open notes, open books. You should turn in the test at the beginning of class on June Quickie Probability Questions Just Write Your Answer. (a) If X is a continuous random variable with p.d.f. f(x) = 4x 3 for 0 < x < c, find c. (b) If X has p.d.f. f(x) = 2x for 0 < x < 1, find E[X 2 ]. (c) If X Exp(λ) represents the lifetime of a lightbulb, find the probability that the lightbulb will survive at least 1/λ time units. (d) If X Exp(λ) represents the lifetime of a lightbulb, find the probability that the lightbulb will survive at least 2/λ time units given that it has already survived at least 1/λ time units. (e) If X 1, X 2,..., X n are i.i.d. Exp(λ), name the distribution of Y = n i=1 X i. (f) TRUE or FALSE? If U Unif(0, 1) then ln(u) has the same distribution as ln(1 U).

2 2 (g) Suppose X, Y, and Z are i.i.d. Bern(0.6). X + Y + Z 2? What is the probability that (h) Suppose X, Y, and Z are i.i.d. Bern(0.6). What is E[X Y Z]? (i) Suppose X = 5 or 7, each with probability 0.5. Find Var(X). (j) Let X and Y be i.i.d. Exponential with rate 1. Find P{X + Y 1}. (k) Suppose X is Normal(3,4). Find P{X > 3}. (l) Suppose X and Y are i.i.d. Normal( 3, 4). Find P{X > Y }. (m) Suppose X 1,..., X 50 are i.i.d. Exp(1/2). What is the approximate distribution of the sample mean, 50 i=1 X i /50? (n) Consider a Poisson arrival process with rate 4/hour. What is the probability that we will observe 10 arrivals in the next 3 hours?

3 3 (o) Consider a Poisson arrival process with rate λ = 1. What is the probability that the time between the 3rd and 4th arrivals will be less than 1? (p) Bonus Question: known as? Who are the 1950 s singing duo Tom and Jerry better

4 4 2. Quickie Arena Questions Just Write Your Answer. (a) TRUE or FALSE? The SEIZE - DELAY - RELEASE blocks are all found in the Advanced Process template. (b) TRUE or FALSE? You can use the RECORD block to capture Tally statistics. (c) TRUE or FALSE? interarrival times. You can use the CREATE block to generate negative (d) TRUE or FALSE? Arena is primarily an event-scheduling modeling language. (e) Show how to use the DISC function to generate a RV that equals 2 with probability 0.3, 2.5 w.p. 0.4., and 4 w.p (f) TRUE or FALSE? You can have customers from two different CREATE blocks feeding into the same PROCESS block. (g) TRUE or FALSE? Two different customers can have two different values for the same attribute.

5 5 (h) TRUE or FALSE? In Arena, a variable is global. (i) In Arena, what is the variance of a NORM(6,8) random variable? (j) TRUE or FALSE? Arena has a future event list. (k) Bonus Question: AC/DC is closely related to what other famous Australian band?

6 6 3. Spectators at a Britney Spears concert have to get two things done in order to attend the concert: (i) Go to the ticket window to purchase their tickets, and (ii) Then go to the metal detection area to be checked for firearms. Note that a concert-goer may have to stand in line at one or both of the stations. Suppose that the first concert-goer arrives at the ticket window at time 0. The next five concert-goers arrive at the ticket window with the following interarrival times: The corresponding six service times at the ticket window are The concert-goers immediately proceed to the firearms detection area. service times at this station are The six (a) How many concert-goers experience a wait in either the ticket queue or firearms detection queue? (b) When does the sixth concert-goer complete service at the firearms station? (c) What is the average time-in-system for the six concert-goers?

7 7 4. Suppose we re interested in evaluating the integral I = 2 1 x2 dx using Monte Carlo integration. (a) Use the following n = 5 Unif(0, 1) random numbers to come up with the usual estimator În for I: (b) What is the expected value of În? (c) If U Unif(0, 1), find Cov(U, 1 U). (d) Re-do Question 4a using the alternative antithetic estimator (where I freely use the obvious notation introduced in class), Ĩ n b a n n f(a + (b a)(1 U i )). i=1 (e) Would you expect În and Ĩn to positively or negatively correlated? (f) Combine your answers in Questions 4a and 4d in a smart way.

8 8 5. Suppose an auto emissions testing center has one server who can perform a test in exactly 10 minutes. Let s assume that arrivals to the center occur according to a Poisson process. What is the largest arrival rate that can be tolerated if the steady-state of the mean queue length is not to exceed 3? 6. Consider the random numbers arising from our favorite generator X i+1 = 16807X i mod(2 31 1). Conduct a χ 2 goodness-of-fit test to test that the numbers are approximately U(0, 1). Use at least 100,000 observations and k = 20 cells. Pick your favorite value of α.