Question

Consider a large 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. Cars and buses are accommodated on different levels of the ferry, so the number of buses accommodated on any trip is independent of the number of cars on the trip. Suppose that x and y have the probability distributions shown below:

x	0	1	2	3	4	5
p(x)	0.04	0.11	0.24	0.31	0.21	0.09

y	0	1	2
p(y)	0.40	0.20	0.40

(a) Compute the mean and standard deviation of x. (Round the answers to three decimal places.)
Mean of x
Standard deviation of x

(b) Compute the mean and standard deviation of y. (Round the answers to three decimal places.)
Mean of y
Standard deviation of y

(c) Compute the mean and variance of the total amount of money collected in tolls from cars. (Round the answers to two decimal places.)
Mean of the total amount of money collected in tolls from cars $
Variance of the total amount of money collected in tolls from cars

(d) Compute the mean and variance of the total amount of money collected in tolls from buses. (Round the answers to one decimal place.)
Mean of the total amount of money collected in tolls from buses $
Variance of the total amount of money collected in tolls from buses

(e) Compute the mean and variance of z = total number of vehicles (cars and buses) on the ferry. (Round the answers to two decimal places.)
Mean of z
Variance of z

(f) Compute the mean and variance of w = total amount of money collected in tolls. (Round the answers to one decimal place.)
Mean of w $
Variance of w

Answer 1

(a) Compute the mean and standard deviation of x. (Round the answers to three decimal places.)
Mean of x
Standard deviation of x

Mean of x : E(X)

Standard deviation of x =

Variance of X = E(X²) - E(X)²

Variance of X = E(X²) - E(X)² = 9.47-2.81² = 9.47 - 7.8961=1.5739

Standard deviation of x =

Mean of x = 2.81
Standard deviation of x = 1.255

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(b) Compute the mean and standard deviation of y. (Round the answers to three decimal places.)
Mean of y
Standard deviation of y

Standard deviation of Y =

Variance of Y = E(Y²) - E(Y)²

Variance of Y = E(Y²) - E(Y)² = 1.8 - 1.0² =1.8-1=0.8

Standard deviation of Y =

Mean of y = 1.0
Standard deviation of y=0.894

(c) Compute the mean and variance of the total amount of money collected in tolls from cars. (Round the answers to two decimal places.)
Mean of the total amount of money collected in tolls from cars $
Variance of the total amount of money collected in tolls from cars

The toll for cars is $3

Mean total amount of money collected in tolls from cars $ = 3 x mean number of cars = 3 x mean of x = 3 x 2.81 = 8.43

Variance of the total amount of money collected in tolls from cars

= 3² x Variance of number of cars = 9 x variance of x = 9 x 1.5739 = 14.1651

Mean total amount of money collected in tolls from cars $ = 8.43

Variance of the total amount of money collected in tolls from cars = 14.1651

(d) Compute the mean and variance of the total amount of money collected in tolls from buses. (Round the answers to one decimal place.)
Mean of the total amount of money collected in tolls from buses $
Variance of the total amount of money collected in tolls from buses

Toll for buses = $10

Mean of the total amount of money collected in tolls from buses $ = 10 x Mean number of buses = 10 x mean of y = 10 x 1=$10

Variance of the total amount of money collected in tolls from buses = 10² x variance of number of buses = 100 x variance of y = 100 x 0.8 =80

Mean of the total amount of money collected in tolls from buses $ = $10
Variance of the total amount of money collected in tolls from buses = 80

(e) Compute the mean and variance of z = total number of vehicles (cars and buses) on the ferry. (Round the answers to two decimal places.)
Mean of z
Variance of z

z = total number of vehicles (cars and buses) on the ferry = Total number of cars on the ferry + Total number of buses on the ferry = x+y

Mean of z = mean of x + mean of y = 2.81+1 =3.81

Variance of z = variance of x + variance of y = 1.5739+0.8=2.3739

Mean of z = 3.81
Variance of z = 2.3739

(f) Compute the mean and variance of w = total amount of money collected in tolls. (Round the answers to one decimal place.)
Mean of w $
Variance of w

w = total amount of money collected in tolls = total amount of money collected in tolls from cars $ + total amount of money collected in tolls from buses $

Mean of w = Mean of  total amount of money collected in tolls from cars $ + mean of  total amount of money collected in tolls from buses $

From (c)

Mean total amount of money collected in tolls from cars $ = 8.43

Variance of the total amount of money collected in tolls from cars = 14.1651

From (d)

Mean of the total amount of money collected in tolls from buses $ = $10
Variance of the total amount of money collected in tolls from buses = 80

Mean of w = Mean of  total amount of money collected in tolls from cars $ + mean of  total amount of money collected in tolls from buses $ = 8.43+10 =18.43

Variance of w = Variance of  total amount of money collected in tolls from cars + Variance of  total amount of money collected in tolls from buses = 14.1651+80=94.1651

Mean of w = 18.43

Variance of w = 94.1651