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Suppose the arrivals of clients to a bank can be modeled as a Poisson process with a given hourly rate λλ . Then the number of clients arriving between 10:00311:00 am is independent of the number of clients arriving between 1:0032:00 pm. Correct! True False (1+e3)/2(1+e3)/2 Correct! 1+(e3/2)1+(e3/2) e3e3 2e32e3 Question 7 3 / 3 pts If X has a mean of 5 and a variance of 2, find Var(20 − 4X). -4 4 8 16 Correct! 32 Question 8 3 / 3 pts Suppose that X is a continuous random variable with p.d.f.f(x)=x2for0x2.FindP(X1|X3/2).Suppose that X is a continuous random variable with p.d.f.f(x)=x2for0x2.FindP(X1|X3/2). 0 1/16 0.2 1/3 Correct! 4/9 1/10 Correct! 1/4 1/2 9/10 Question 10 3 / 3 pts Suppose that X and Y have joint p.d.f.f(x,y)=cxy2/(1+x3+y3),for0≤x≤1and1≤y≤3,where c is whatever constant makes this thing integrate to 1. Are X and Y independent ? YES or NO ?Suppose that X and Y have joint p.d.f.f(x,y)=cxy2/(1+x3+y3),for0≤x≤1and1≤y≤3,where c is whatever constant makes this thing integrate to 1. Are X and Y independent ? YES or NO ? YES Correct! NO I=+32e212(x22)2dxI=∫23e−12(x−2)2dx Use the following four Unif(0, 1) random numbers to compute a Monte Carlo estimate of I: 0.13 0.35 0.64 0.87 Here’s the formula from class that ought to be used to calculate the estimate: In^=b2an∑ni=1g(a+(b2a)Ui)In^=b−an∑i=1ng(a+(b−a)Ui) 0.134 0.342 0.456 Correct! 0.858 0.996 Question 12 3 / 3 pts Toss 1000 darts uniformly into a unit square. In class, we saw that we can use the proportion p^p^ of darts that land in the circle of radius 1/2 inscribed in that unit square to get an estimate of ππ . If 781 of the darts fall in the circle, what is our method’s estimate of ππ ? 3.000 3.082 Correct! 3.124 3.166 1.08 ππ Question 13 3 / 3 pts Suppose U and V are i.i.d. Unif(0,1) random variables. Select from the options below an appropriate expression to generate a realization of the sum of two 10-sided dice tosses. (Recall that +x,+x, is the ceiling= or round up= function.) Correct! (A) +10U,++10V,+10U,++10V, (B) (0.5∗+20U,)+(0.5∗+20V,)(0.5∗+20U,)+(0.5∗+20V,) (C)+20U,+20U, (d) All of the above Only (a) and (c) True or False ? Discrete-event simulations are particularly suitable for analyzing continuous-flow phenomena such as the velocity and altitude of an aircraft as it comes in for a landing. True Correct! False Question 2 3 / 3 pts Consider a single-server queue simulation with i.i.d. exponential interarrivals, i.i.d. exponential services, and a first-in-first-out service discipline. Suppose that the arrival rate is 5 per hour, and the service rate is 4 per hour. What will happen in the long run? (a) The server will be busy about 20% of the time. Correct! (b) The server will be busy all of the time. (c) The number of custmers in the system (line + in service) will never exceed 9. (d) The length of the line will never exceed 5. (e) The average number of people in the queue will converge to around 4. False Question 4 3 / 3 pts The planet Glubnor has 120-day years. Suppose there are four Glubnorians in the room. What is the probability that at least two of them share the same birthday? 0.015 0.025 Correct! 0.049 0.149 0.975 Question 5 3 / 3 pts Suppose the arrivals of clients to a bank can be modeled as a Poisson process with a given hourly rate λλ . Then the number of clients arriving between 10:00311:00 am is independent of the number of clients arriving between 1:0032:00 pm. Correct!