ISYE 6644 EXAM 1 |COMPLETE QUESTIONS WITH 100% GRADED EXPERT
SOLUTIONS | 100% CORRECT | GET A+
1. 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.: False
2. 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?: b) The server will be busy all of the
time.
3. If the covariance of X and Y is 1/2, then X and Y cannot be independent.: True 4. 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?: 0.049
5. 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 10:00-11:00 am is independent of the
number of clients arriving between 1:00-2:00 pm.: True
6. IfXBern(0.5),findE[12+e3X]IfX< Bern(0.5),findE[12+e3X]< : 1+(e3/2)1+(e3/2)
7. If X has a mean of 5 and a variance of 2, find Var(20 − 4X).: 32
8. Suppose that X is a continuous random variable with p.d.f.f(x)=x^2 for 0
<X<2. Find P(X<1 | X < 3/2): 4/9
9. Suppose that X and Y are i.i.d. with a mean of −5, a variance of 4, and Cov(X, Y) = 1. Find
Corr(X, Y).: 1/4
1/9
, 10. Suppose that X and Y have joint p.d.f.f(x,y)=cxy2/(1+x3+y3),for0xd1dand1dy3d,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),for0xd1dand1yd3d,where c is whatever constant makes this thing integrate
to 1. Are X and Y independent ? YES or NO ?: NO
11. Consider the following integral: I=32+ e−12(x−2)2dxI=23+ e−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^=b−anni=1g(a+(b−a)Ui)In^=b−an i=1ng(a+(b−a)Ui) : 0.858
12 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 ?: 3.124
13. 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.): (A) 10 U+ 10 V10 U+ 10 V
14. Consider the discrete random vari-
able X: P(X=x)=0.1§©¨ªªªª,ifx=−3.70.4,ifx=20.5,ifx=60,otherwiseP(X=x)
={0.1,ifx=−3.70.4,ifx=20.5,ifx=60,otherwise Use the discrete version of the Inverse Transform
method from class and the Unif(0, 1) pseudo-random number U = 0.13 to generate one observation X
coming from this p.m.f.: X = 2 15. Consider the linear congruential
2/9
SOLUTIONS | 100% CORRECT | GET A+
1. 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.: False
2. 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?: b) The server will be busy all of the
time.
3. If the covariance of X and Y is 1/2, then X and Y cannot be independent.: True 4. 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?: 0.049
5. 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 10:00-11:00 am is independent of the
number of clients arriving between 1:00-2:00 pm.: True
6. IfXBern(0.5),findE[12+e3X]IfX< Bern(0.5),findE[12+e3X]< : 1+(e3/2)1+(e3/2)
7. If X has a mean of 5 and a variance of 2, find Var(20 − 4X).: 32
8. Suppose that X is a continuous random variable with p.d.f.f(x)=x^2 for 0
<X<2. Find P(X<1 | X < 3/2): 4/9
9. Suppose that X and Y are i.i.d. with a mean of −5, a variance of 4, and Cov(X, Y) = 1. Find
Corr(X, Y).: 1/4
1/9
, 10. Suppose that X and Y have joint p.d.f.f(x,y)=cxy2/(1+x3+y3),for0xd1dand1dy3d,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),for0xd1dand1yd3d,where c is whatever constant makes this thing integrate
to 1. Are X and Y independent ? YES or NO ?: NO
11. Consider the following integral: I=32+ e−12(x−2)2dxI=23+ e−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^=b−anni=1g(a+(b−a)Ui)In^=b−an i=1ng(a+(b−a)Ui) : 0.858
12 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 ?: 3.124
13. 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.): (A) 10 U+ 10 V10 U+ 10 V
14. Consider the discrete random vari-
able X: P(X=x)=0.1§©¨ªªªª,ifx=−3.70.4,ifx=20.5,ifx=60,otherwiseP(X=x)
={0.1,ifx=−3.70.4,ifx=20.5,ifx=60,otherwise Use the discrete version of the Inverse Transform
method from class and the Unif(0, 1) pseudo-random number U = 0.13 to generate one observation X
coming from this p.m.f.: X = 2 15. Consider the linear congruential
2/9
