91Ó°ÊÓ

The given information is:

If the customer died at the age of 25, the company had to pay out the face value of $100,000 because the client had paid the premium five times in total (as the client had paid the premium yearly since he was 21).

The company's profit was then calculated as the total of the 5 premiums (250 each premium) minus the cost of 100,000. Profit:

$$\begin{gathered} =5x\$250-\$100,000 \\ =\$1,250-\$100,000 \\ =-\$98,750 \end{gathered}$$

$$\begin{gathered} \mathrm{μ}=∑xP\left(X=x\right) \\ =\left(-99,750\right)×\left(0.00183\right)+\left(-99,500\right)×\left(0.00186\right)+\left(-99,250\right)×\left(0.00189\right)+ \\ \left(-99,000\right)×\left(0.00191\right)+\left(98,750\right)×\left(0.00193\right)+\left(1250\right)×\left(0.995058\right) \\ =303.3525 \end{gathered}$$

The average profit on the randomly selected policy of the firm is $303.3525

The given mean is 303.3525.

The predicted value of the squared departure from the mean is known as the variance:

$$\begin{gathered} {\mathrm{σ}}^2=∑{\left(x-\mathrm{μ}\right)}^{2}P\left(x\right) \\ ={\left(-99750-303.3525\right)}^2×\left(0.00183\right)+{\left(-99500-303.3525\right)}^2×\left(0.00186\right)+ \\ {\left(99250-303.3525\right)}^2×\left(0.00189\right)+{\left(99000-303.3525\right)}^2×\left(0.00191\right)+ \\ {\left(98750-303.3525\right)}^2×\left(0.00193\right)+{\left(1250-303.3525\right)}^2×\left(0.99058\right) \\ ≈94,236,827 \end{gathered}$$

The standard deviation is:

$$\begin{gathered} \mathrm{σ}=\sqrt{{\mathrm{σ}}^2} \\ =\sqrt{94,236,827} \\ ≈9707.57 \end{gathered}$$