Question

Rotter Partners is planning a major investment. The amount of profit $X$(in millions of dollars) is uncertain, but an estimate gives the following probability distribution:

Based on this estimate, ${\mu }_{\mathrm{X}}=3$and ${\sigma }_{\mathrm{X}}=2.52$

Rotter Partners owes its lender a fee of $\$200,000$plus $10\%$of the profits $X$. So the firm actually retains $Y$$=0.9X-0.2$from the investment. Find the mean and standard deviation of the amount $Y$that the firm actually retains from the investment.

Short answer

The mean and standard deviation are

${\mu }_Y=2.5\text{million dollars}$

${\sigma }_Y=2.268\text{million dollars}$

Step-by-step solution

Given Information

Given in the question that

$$\begin{gathered} {\mu }_{\mathrm{X}}=3 \\ {\sigma }_{\mathrm{X}}=2.52 \\ Y=0.9X-0.2 \end{gathered}$$

Explanation

Properties mean and standard deviation

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

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

Then we obtain for $Y=0.9X-0.2$

localid="1649908061806" $$\begin{gathered} {\mu }_Y={\mu }_{0.9X-0.2} \\ =0.9{\mu }_X-0.2 \\ =0.9\left(3\right)-0.2 \\ =2.5 \end{gathered}$$

localid="1649908064829" $$\begin{gathered} {\sigma }_Y={\sigma }_{0.9X-0.2} \\ =0.9{\sigma }_X \\ =0.9(2.52) \\ =2.268 \end{gathered}$$