ISYE 6644 2 COMPREHENSIVE STUDY
GUIDE 2026 FULL QUESTIONS AND
SOLUTIONS GRADED A+
◍ 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..
Answer: False
◍ 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?.
Answer: b) The server will be busy all of the time.
◍ If the covariance of X and Y is 1/2, then X and Y cannot be independent..
Answer: True
◍ 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?.
Answer: 0.049
◍ 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..
Answer: True
◍ IfX Bern(0.5),findE[12+e3X]IfX Bern(0.5),findE[12+e3X].
Answer: 1+(e3/2)1+(e3/2)
, ◍ If X has a mean of 5 and a variance of 2, find Var(20 − 4X)..
Answer: 32
◍ 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).
Answer: 4/9
◍ 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)..
Answer: 1/4
◍ 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 ?.
Answer: NO
◍ Consider the following integral: I=∫32e−12(x−2)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^=b−an∑ni=1g(a+(b−a)Ui)In^=b−an∑i=1ng(a+(b−a)Ui).
Answer: 0.858
◍ 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 ?.
Answer: 3.124
◍ 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
GUIDE 2026 FULL QUESTIONS AND
SOLUTIONS GRADED A+
◍ 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..
Answer: False
◍ 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?.
Answer: b) The server will be busy all of the time.
◍ If the covariance of X and Y is 1/2, then X and Y cannot be independent..
Answer: True
◍ 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?.
Answer: 0.049
◍ 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..
Answer: True
◍ IfX Bern(0.5),findE[12+e3X]IfX Bern(0.5),findE[12+e3X].
Answer: 1+(e3/2)1+(e3/2)
, ◍ If X has a mean of 5 and a variance of 2, find Var(20 − 4X)..
Answer: 32
◍ 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).
Answer: 4/9
◍ 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)..
Answer: 1/4
◍ 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 ?.
Answer: NO
◍ Consider the following integral: I=∫32e−12(x−2)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^=b−an∑ni=1g(a+(b−a)Ui)In^=b−an∑i=1ng(a+(b−a)Ui).
Answer: 0.858
◍ 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 ?.
Answer: 3.124
◍ 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
