2024_ISYE-6644-Mid2_Notes

Find E[2X+ 1].Suppose X and Y are continuous RV with joint p.d.f. f(x, y) = cxy^2,0≤x≤2,0≤y≤1, for some appropriate constant c 11/3 Find Cov(2X, −3Y+ 2). If X and Y are independent Nor(0,1) RV 0 Find Var(X+Y). Suppose X and Y are discrete RV with the following joint p.m.f 0.49 What is the distribution given by the Arena expression UNIF(1,2)+UNIF(1,2)? Tria(2,3,4) If U1 and U2 are i.i.d. Unif(0,1), find the distribution of U1−U2 Tria(−1,0,1) Joey’s Lightbulb Company sells bulbs that have Exp(λ= 1/year) lifetimes. Sup-pose I just bought 100 lightbulbs from Joey. What is the approximate probability that the average of the 100 bulb lifetimes will be at least 1.1 years 0.1587 OR 0.16 Suppose X is a discrete RV that can take on four possible values: −0.1, 0, 2, and 4. The associated probabilities are, respectively: 0.3, 0.1, 0.2, and 0.4. Find an expression in Arena to generate one observation coming from this p.m.f. DISC(0.3, −0.1 , 0.4, 0, 0.6, 2, 1, 4) Suppose there are 3 people in the line called joe.queue and 5 people in the line called tom.queue. What is the value of the following Arena expression? (NQ(joe.queue) > NQ(tom.queue)) + (TNOW >= 0) False(0) + True(1) ==> 1 Suppose there are 7 people in the line called joe.queue and 2 people in the line called tom.queue. What is the value of the following Arena expression? (NQ(tom.queue) >= NQ(joe.queue)) + (TNOW >−1) False(0) + True(1) ==> 1 Consider the full-period linear congruential generator Xi+1 = (5Xi+ 3)mod(16).Using X0= 0, calculate the 49th intege X49 3 Regarding Linear Congruential Generators (LCG’s) Both full periodicity and cycle length properties should be considered when choosing a good LCG Consider our desert island generator Xi+1 = 16807 Xi mod(2^31 −1). If X0=123456789, find the value of the first PRN , U1 0.218 Consider our desert island generator Xi+1 = 16807 Xi mod(2^31 −1). If X0=1234321, find the value of X1 1,417,880,224 Suppose that a Tausworthe generator gave you the series of bits 11011. If you useall of these bits, what Unif(0,1) random number would that translate to? 0.8844 Consider the PRN generator Xi+1 = (aXi+ 5)mod(b). Using X0= 0, which values of 'a' and 'b' yield a cycle length of 4? a=3 & b=8 Suppose U1, U2, U3 are i.i.d. Unif(0,1). What is the probability that we will seeexactly 1 run up and down? 1/3 Consider the following sequence of PRN’s (0.98|0.95|0.23|0.17|0.12|0.03|0.47|0.83|0.63|0.72) 4 runs Suppose that we have a sequence of n= 100 PRN’s , and we observe 80 runs up and down. Using α= 0.01, do we ACCEPT (i.e., fail to reject) or REJECT the null hypothesis of independence? (2.57) Reject H0 We are testing a new medication. Our null hypothesis , H0, is the assumption that the medication is safe. If we reject this null hypothesis but the medication is actually safe, what type of error have we committed? Type I Error Suppose you are doing a X^2 goodness-of-fit test with level of significance α= 0.01 and k= 5 cells. Further suppose that there are no parameters to estimate so that your g-o-f statistic X^2|0 has k−1 degrees of freedom. If it happens that X^2|0= 36.00, do we ACCEPT (i.e., fail to reject) or REJECT the null hypothesis of uniformity? REJECT uniformity Consider 36 PRNs , U1, U2, ..., U36, and suppose that SUM<36|i=1> Ui= 20.8. Use these PRNs to generate a single approximately Nor(0,1) RV via our “desert island” technique. 1.617 Suppose U1= 0.12, U2= 0.93, U3= 0.50, U4= 0.11, and U5= 0.49. Use these values and apply the Acceptance- Rejection technique to generate a realization of a Pois(λ= 2.4) RV. (You may not need to use all of the PRN’s.) N=2 Which of the following is NOT a method used to generate RV? Computation- Compilation Method What is (0 XOR 1) XOR (1 XOR 1)? 1 Consider a Tausworthe generator with r= 1, q= 3, B1= 1, B2= 1, and B3= 1. Find B39 0 Suppose that we have a sequence of n= 100 PRN’s , and we observe 80 runs upand down. Using α= 0.05, do we ACCEPT (i.e., fail to reject) or REJECT the null hypothesis of independence? REJECT H0. In other words, we conclude that they’re not

independent. Suppose the RV 'X' has p.d.f. f(x) = 2(x−1) for 1≤x≤2. Find the inverse of its c.d.f., i.e., F−1(U), where U is a PRN U^(1/2) + 1  Let’s toss a pair of dice and look at the sum. The number of trials Xit takes until I see a sum of 4 is easily shown to be a Geom(1/12) RV. Use the PRNU= 0.5 to generate X via inverse transform. 8 Let’s toss a pair of dice and look at the sum. The number of trials X it takes until I see a sum of 2 is easily shown to be a Geom(1/36) RV. Usethe PRN U= 0.75 to generate X via inverse transform? Opt 1==> X=[(ln(1-U)/ln(1-P))] = [ln(1-0.75)/ln(35/36)]= 49.12 ==> 50 Opt 2==> [ln(U)/ln(1 −p)]=[ln(0.75)/ln(35/36)]= 11 50 -OR - 11 Consider 48 PRNs, U1, U2, ..., U48 , and suppose that SUM<48|i=1> Ui= 24.8. Use these PRNs to generate a single approximately Nor(0,1) RV via our “desert island” technique 0.4 Suppose that U1 and U2 are i.i.d. Unif(0,1). Find Pr (ln(U1 U2)>−2) 0.594 What is the expected value of the Arena expression EXPO(DISC(0.5,2,1,3))? 2.5 In Arena, what kind of module would you use to change an entity’s picture? ASSIGN In Arena, where would you find the Expression spreadsheet? Advanced Process panel Consider the Arena Inventory model that we did in class as a demo. Did this model allow for backlogs? Y/N? Yes What is the name of Arena's primary modeling approach "world view"? Process Interaction What sequence of Arena modules would you use to generate customer arrivals, use a server, and then have customers leave the system after they're done with the server? Create-Process- Dispose An ENTER module is usually paired with what other module? LEAVE Consider von Neumann's  mid -square PRN method, and suppose that X0= 6632.  What is R3 =X3/10000? [Note: For purposes of this problem, treat all Xi's as if they were 8 digits, e.g., treat 123456 as if it were 00123456.] 0.0556 Suppose the guy on trial is actually guilty but you incorrectly acquit him. So you've incorrectly accepted the null hypothesis of innocence. What type of error have you just made - Type I or Type II? Type II Unif(0,1) PRNs can be used to generate which of the following random entities? All, Exp(λ), Nor(0,1), triaguar, Bern(p) NonHomo.. expected If X is an Exp(λ) RV with c.d.f. F(x)=1-e^λX, what's the distribution of the RV 1-e^λX? Unif(0,1) If U is a Unif (0,1) RV, what's the distribution of -(1/λ)ln(U)? Exp(λ) How would you simulate the sum of two 6-sided dice tosses? (Note that [.] is the round-up function; and all of the U's denote PRNs.) [6U1] + [6U2] Let W(t) denote a Brownian motion process at time 't'. Calculate Cov(W(3), W(5)). ==> min(3,5) = 3 3 If U is Unif (0,1), how can we simulate a Geom(0.6) random variate? Opt1-> [ln(U)/ln(0.4)], And Opt2-> [ln(1- U)/ln(0.4)]

In Arena, it is possible to set the maximum capacity of a queue from within a SEIZE module. In Arena, the expression NORM (-1,2)+ NORM (3,2) has the same distribution as the expression NORM(2,4) In Arena, it is only possible to use one resource at a time in a PROCESS module. In Arena, once an entity is defined, it is not possible to change the priority in which it gets processed by a resource. In Arena, it is not possible to use a single RELEASE module to release multiple different resources. Most discrete-event simulations proceed by moving the simulation clock to the most-computationally-intensive event on the future eventslist, i.e., the event that takes the “most work” to deal with; executing that event(including any adds, deletes, or swaps to the FEL); and then repeating this cycle There are various PRN generators out there having periods greater than 2^19900, but since those periods are so big we can’t really use such generators at all In Arena, a primitive QUEUE block can connect with a PROCESS module. In Arena, you are only allowed to schedule one type of failure for a particular resource. All of the Tech Support servers for our Arena Call Center simulation are equally skilled and have the same concurrent work schedules. A Sequence must consist of unique visitation locations... no repeat visits! You SEIZE-DELAY-RELEASE a conveyor.   In general, the majorizing function t(x) is itself a p.d.f. f(x). Is it ever OK to use the midSquare random number generator? NO! The generator Xi = (32Xi−1 + 124)mod(2048) has full period Most discrete-event simulations proceed by moving the simulation clock to the most-computationally-intensive event on the future events list, i.e., the event that takes the “most work” to deal with; executing that event (including any adds, deletes, or swaps to the FEL); and then repeating this cycle There are various PRN generators out there having periods greater than 2^19900, but since those periods are so big we can’t really use such generators at all In Arena, what kind of module would you use to give a new value to an attribute? ASSIGN What is the name of the Arena module that can be used to split one customer into two or more clones? Separate of clones What does “LCG” mean? Linear Congruential Generator