An insurance company’s monthly claims are modeled by a continuous, positive random variable [math]X[/math], whose probability density function is proportional to [math](1 + x)^{- 4}[/math] for [math] x \gt 0 [/math] .

Calculate the company’s expected monthly claims.

• 1/6
• 1/3
• 1/2
• 1
• 3

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

Solution: C

The pdf of x is given by [math]f(x) = \frac{k}{(1+x)^4}, \, 0 \lt x \lt \infty [/math]. To find [math]k[/math], note

[[math]] 1 = \int_0^{\infty} \frac{k}{(1+x)^4} dx = - \frac{k}{3} \frac{1}{(1+x)^3} \Big |_{0}^{\infty} = \frac{k}{3} [[/math]]

which implies that [math]k = 3 [/math]. It then follows that

[[math]] \operatorname{E}[x] = \int_0^{\infty} \frac{3x}{(1+x)^4} dx [[/math]]

and substituting [math]u = 1+x, du = dx [/math], we see

[[math]] \begin{align*} \operatorname{E}[x] = \int_1^{\infty} \frac{3(u-1)}{u^4} du &= 3 \int_{1}^{\infty} (u^{-3} - u^{-4}) du \\ &= 3\left[\frac{u^{-2}}{-2} - \frac{u^{-3}}{-3}\right]_0^{\infty} \\ &= 3\left[\frac{1}{2}-\frac{1}{3}\right] = 3/2 - 1 = 1/2. \end{align*} [[/math]]

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