Let X be the damage incurred (in $) in a certain type of accident during a given year. Possible X values are 0, 1000, 5000, and 10000, with probabilities 0.81, 0.09, 0.08, and 0.02, respectively. A particular company offers a $500 deductible policy. If the company wishes its expected profit to be $100, what premium amount should it charge?

Let the random variable [tex]X[/tex] denote the damage in $ incurred in a certain type of accident during a given year. The probability distribution of [tex]X[/tex] is given by

[tex]X : \begin{pmatrix}0 & 1000 & 5000 & 10\; 000\\0.81 & 0.09 & 0.08 & 0.02\end{pmatrix}[/tex]

A company offers a $500 deductible policy and it wishes its expected profit to be $100. The premium function is given by

[tex]F(x) = \left \{ {{X+100, \quad \quad \quad \quad \quad \text{for} \; X = 0 } \atop {X-500+100} , \quad \text{for} \; X = 500,4500,9500} \right.[/tex]

For [tex]X = 0[/tex], we have

[tex]F(X) = 0+100 = 100[/tex]

For [tex]X = 500[/tex],

[tex]F(X) = 500-500+100 = 100[/tex]

For [tex]X = 4500[/tex],

[tex]F(X) = 4500-500+100 = 4100[/tex]

For [tex]X = 9500[/tex],

[tex]F(X) = 9500-500+100 = 9100[/tex]

Therefore, the probability distribution of [tex]F[/tex] is given by

[tex]F : \begin{pmatrix} 100 & 100 & 41000 & 91000\\0.81 & 0.09 & 0.08 & 0.02\end{pmatrix}[/tex]

To determine the premium amount that the company should charge, we need to calculate the expected value of [tex]F.[/tex]

[tex]E(F(X)) = \sum \limits_{i=1}^{4} f(x_i) \cdot p_i = 100 \cdot 0.81 + 100 \cdot 0.09 + 4100 \cdot 0.08 + 9100 \cdot 0.02[/tex]

Therefore,

[tex]E(F) = 81+9+328+182 = 600[/tex]

which means the $600 is the amount the should be charged.