Question

Refer to Exercise $37$. The ferry company’s expenses are $\$20$per trip. Define the random variable $Y$to be the amount of profit (money collected minus expenses) made by the ferry company on a randomly selected trip. That is, $Y=M-20$.

(a) How does the mean of $Y$relate to the mean of $M$? Justify your answer. What is the practical importance of ${\mu }_Y$?

(b) How does the standard deviation of $Y$relate to the standard deviation of $M$? Justify your answer. What is the practical importance of ${\sigma }_Y$?

Short answer

(a)${\mu }_Y={\mu }_M-20$

The expected loss per trip is $\$0.65$.

(b)${\sigma }_Y={\sigma }_M=6.45$

The expected costs are expected to vary by about $\$6.45$from the expected loss pf- $\$0.65$per trip.

Step-by-step solution

Part (a) Step 1: Given Information 

Given:

$Y=M-20$

Part (a) Step 2: Explanation 

Result of previous exercise:

$$\begin{gathered} {\mu }_M=19.35 \\ {\sigma }_M=6.45 \end{gathered}$$

Property mean:

${\mu }_{aX+b}=a{\mu }_X+b$

Then we can determine the mean for $Y$if we know the mean of $M$:

$$\begin{gathered} {\mu }_Y={\mu }_{M-20} \\ ={\mu }_M-20 \\ =19.35-20 \\ =-0.65 \end{gathered}$$.

Part (a) Step 1: Given Information 

Given:

$Y=M-20$

Part (b) Step 2: Explanation 

Property standard deviation:

${\sigma }_{aX+b}=a{\sigma }_X$

Result of the previous exercise:

$$\begin{gathered} {\mu }_M=19.35 \\ {\sigma }_M=6.45 \end{gathered}$$

Then we can determine the standard deviation for $Y$:

$$\begin{gathered} {\sigma }_Y={\sigma }_{M-20} \\ ={\sigma }_M \\ =6.45 \end{gathered}$$

The practical importance is that the fixed costs do not influence the spread of the distribution of money (profit/money collected).