Question

A car dealership sells 0, 1, or 2 luxury cars on any day. When
selling a car, the dealer also tries to persuade the customer to
buy an extended warranty for the car. Let X denote the number of
luxury cars sold in a given day, and let Y denote the number of
extended warranties sold.

P(X = 0,Y = 0) = 1/6;

P(X = 1,Y = 0) = 1/12;

P(X = 1,Y = 1) = 1/6;

P(X = 2,Y = 0) = 1/12;

P(X = 2,Y = 1) = 1/3;

P(X = 2,Y = 2) = 1/6.

(a) Find the marginal distributions of X and Y; (Drawing the
table of the joint distribution might be helpful.)

(b) Calculate conditional distribution p (x | y = 0) and E (X | Y =
0);

(c) Calculate Cov(X, Y ) and the correlation of X and Y, ρXY. Are X
and Y independent random variables? Explain the reason.

(d) Calculate Va r (X + Y ).

Answer #1

a) The marginal distributions are,

b) The conditional distribution is

The conditional expectation is

c) The Covariance is found as

Since , X, Y are not independent.

d) The variance

A car dealership (which is opened 7 days a week) sells an
average of 4 cars in a day. Assume the number of cars sold each day
is independent from any other day. The number of cars sold on any
given day can be approximated with a Poisson distribution. Find the
probability that the car dealership will sell 6 cars tomorrow.
A) 28
B) 0.1107
C) 0.8958
D)4
E) 0.8893
F) 0.1042

Suppose there are only two types of cars in the used car market
q=0 and q=1. Half the cars are q=0 and the other half are q=1.
Buyers still cannot tell the quality but they are aware of the
quality distribution. Sellers are willing to accept any price p ?
0, but prefer to receive a higher price. If buyers do not know q,
then they are willing to pay p=10000*Q+500 where Q is the average
quality of the cars...

A dishonest used-car dealer sells a car to an unsuspecting
buyer, even though the dealer knows that the car will have a major
breakdown within the next 6 months. The dealer provides a warranty
of 45 days (i.e. 1.5 months) on all cars sold. Let x represent the
length of time until the breakdown occurs. Assume that x is a
uniform random variable with values between 0 and 6 months.
a. Calculate and interpret the mean of x.
b. Calculate...

4. Ethel is trying to decide whether to have 0 cars, 1 car, or 2
cars. If x is the number of cars she has and y is the amount of
money she has per year to spend on other stuff, Ethel’s utility
function is U(x, y), where U(0, y) = y^1/2, U(1, y) = (15/14)y^1/2,
and U(2, y) = (10/9)y^1/2. Suppose that it costs $2,000 a year to
have 1 car and $4,000 a year to have 2 cars....

Consider a small ferry that can accommodate cars and buses. The
toll for cars is $3, and the toll for buses is $10. Let X
and Y denote the number of cars and buses, respectively,
carried on a single trip. Suppose the joint distribution of
X and Y is as given in the table below.
y
p(x,y)
0
1
2
x
0
0.025
0.015
0.010
1
0.050
0.030
0.020
2
0.110
0.075
0.050
3
0.150
0.090
0.060
4
0.100
0.060...

Consider a small ferry that can accommodate cars and buses. The
toll for cars is $3, and the toll for buses is $10. Let X
and Y denote the number of cars and buses, respectively,
carried on a single trip. Suppose the joint distribution of
X and Y is as given in the table below.
y
p(x,y)
0
1
2
x
0
0.025
0.015
0.010
1
0.050
0.030
0.020
2
0.050
0.075
0.050
3
0.150
0.090
0.060
4
0.100
0.060...

Consider a small ferry that can accommodate cars and buses. The
toll for cars is $3, and the toll for buses is $10. Let X
and Y denote the number of cars and buses, respectively,
carried on a single trip. Suppose the joint distribution of
X and Y is as given in the table below.
y
p(x,y)
0
1
2
x
0
0.025
0.015
0.010
1
0.050
0.030
0.020
2
0.085
0.075
0.050
3
0.150
0.090
0.060
4
0.100
0.060...

Consider a small ferry that can accommodate cars and buses. The
toll for cars is $3, and the toll for buses is $10. Let X
and Y denote the number of cars and buses, respectively,
carried on a single trip. Suppose the joint distribution of
X and Y is as given in the table below.
y
p(x,y)
0
1
2
x
0
0.025
0.015
0.010
1
0.050
0.030
0.020
2
0.105
0.075
0.050
3
0.150
0.090...

Consider a small ferry that can accommodate cars and buses. The
toll for cars is $3, and the toll for buses is $10. Let X
and Y denote the number of cars and buses, respectively,
carried on a single trip. Suppose the joint distribution of
X and Y is as given in the table below.
y
p(x,y)
0
1
2
x
0
0.025
0.015
0.010
1
0.050
0.030
0.020
2
0.105
0.075
0.050
3
0.150
0.090...

Consider a small ferry that can accommodate cars and buses. The
toll for cars is $3, and the toll for buses is $10. Let X
and Y denote the number of cars and buses, respectively,
carried on a single trip. Suppose the joint distribution of
X and Y is as given in the table below.
y
p(x,y)
0
1
2
x
0
0.025
0.015
0.010
1
0.050
0.030
0.020
2
0.125
0.075
0.050
3
0.150
0.090
0.060
4
0.100
0.060...

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