ISYE 6644 EX 1 exam with correct ||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\
answers
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
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.
||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\
If the covariance of X and Y is 1/2, then X and Y cannot be
||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\
independent.
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?
||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\
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.
||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\
True
IfX∼Bern(0.5),findE[12+e3X]IfX∼Bern(0.5),findE[12+e3X]
1+(e3/2)1+(e3/2)
If X has a mean of 5 and a variance of 2, find Var(20 − 4X).
||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\
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) ||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\
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).
||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\
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 ? ||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\
answers
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
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.
||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\
If the covariance of X and Y is 1/2, then X and Y cannot be
||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\
independent.
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?
||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\
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.
||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\
True
IfX∼Bern(0.5),findE[12+e3X]IfX∼Bern(0.5),findE[12+e3X]
1+(e3/2)1+(e3/2)
If X has a mean of 5 and a variance of 2, find Var(20 − 4X).
||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\
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) ||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\
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).
||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\
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 ? ||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\ ||\\||\\
