Once a fire is reported to a fire insurance company, the company makes an initial estimate, [math]X[/math], of the amount it will pay to the claimant for the fire loss. When the claim is finally settled, the company pays an amount, [math]Y[/math], to the claimant. The company has determined that [math]X[/math] and [math]Y[/math] have the joint density function

[[math]] f(x,y) = \begin{cases} \frac{2}{x^2(x-1)}y^{-(2x-1)/(x-1)}, \,\, x \gt1, y \gt 1 \\ 0, \, \textrm{Otherwise.} \end{cases} [[/math]]

Given that the initial claim estimated by the company is 2, calculate the probability that the final settlement amount is between 1 and 3.

• 1/9
• 2/9
• 1/3
• 2/3
• 8/9

Copyright 2023. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.

Solution: E

[[math]] \begin{align*} \operatorname{P}[1 \lt Y \lt 3 | X = 2] = \int_1^3 \frac{f(2,y)}{f_x(2)} dy \\ f(2,y) = \frac{2}{4(2-1)} y^{-(4-1)/2 -1} = \frac{1}{2}y^{-3} \\ f_x(2) = \int_1^{\infty} \frac{1}{2}y^{-3} dy = - \frac{1}{4}y^{-2} \Big |_0^{\infty} = \frac{1}{4} \end{align*} [[/math]]

Finally,

[[math]] \operatorname{P}[1 \lt Y \lt 3 | X = 2] = \frac{\int_1^3 \frac{1}{2} y^{-3} dy}{\frac{1}{4}} = -y^2 \Big |_1^3 = 1- \frac{1}{9} = \frac{8}{9}. [[/math]]