Question

A small ferry runs every half hour from one side of a large river to the other. The number of cars $X$on a randomly chosen ferry trip has the probability distribution shown here with mean ${\mu }_X=3.87$and standard deviation ${\sigma }_X=1.29$. The cost for the ferry trip is $\$5$. Define $M=$money collected on a randomly selected ferry trip.

a. What shape does the probability distribution of $M$have?

b. Find the mean of $M$.

c. Calculate the standard deviation of $M$.

Short answer

a. The form of $M\text{'}s$distribution is then the same as that of $X\text{'}s$, skewed to the left.

b. Average is $\$19.35$in cash.

c. The amount of money gained on a randomly selected boat voyage varies by $\$6.45$on average from the mean value of $\$19.35$.

Step-by-step solution

Part(b) Step 1 : Given Information 

Probability distribution shown :

Mean : $\mu X=3.87$

Standard deviation : $\sigma X=1.29$

The cost for the ferry trip is :$\$5$

Part(a) Step 2 : Simplification 

$X$is the number of automobiles chosen at random, while $M$denotes the amount of money collected on a randomly chosen ferry journey.

The cost of a ferry journey per car is $\$5$, hence the amount of money gained from the boat trip is the product of the number of automobiles and the cost of the ferry trip per car.

$5\times X=M$

The form of the distribution remains unaffected if every value is multiplied by the same constant.

Because the lowest bar in the given histogram is to the left and the highest bar is to the right, the distribution is skewed to the left.

Part(b) Step 1 : Given Information 

Probability distribution shown :

Mean : $\mu X=3.87$

Standard deviation :$\sigma X=1.29$

The cost for the ferry trip is :$\$5$

Part(b) Step 2 : Simplification 

$$\begin{gathered} M=5\times {\mu }_x \\ =5\times 3.87 \\ =\mathbf{19}\mathbf{.}\mathbf{35} \end{gathered}$$,

where $X$represents the number of cars chosen at random, and $M$represents the money collected on a random ferry trip.

The cost of a ferry journey per car is $\$5$, hence the amount of money gained from the boat trip is the product of the number of automobiles and the cost of the ferry trip per car.

$M=5\times X$

If every value is multiplied by the same constant, the distribution's center is also multiplied by the same constant, resulting in $M\text{'}s$mean being multiplied by $5.$An arbitrary decision will yield an average of $\mathbf{\$}\mathbf{19}\mathbf{.}\mathbf{35}$in cash.

Part(c) Step 1 : Given Information 

Probability distribution shown :

Mean : $\mu X=3.87$

Standard deviation : $\sigma X=1.29$

The cost for the ferry trip is : $\$5$

Part(c) Step 2 : Simplification 

$$\begin{gathered} {\sigma }_M=5\times {\sigma }_x \\ =5\times 1.29 \\ =\mathbf{6}\mathbf{.}\mathbf{45} \end{gathered}$$

where $X$represents the number of cars chosen at random and $M$represents the money received on a random ferry journey.

The cost of a ferry journey per car is $\$5$, hence the money saved on the trip is the product of the number of cars and the cost of a ferry trip per car $M=5\times X$.

When every value is multiplied by the same constant, the spread of the distribution is modified in the same way, resulting in a standard deviation of $5.$