You are given:

Claim Size (X)	Number of Claims
(0,25]	25
(25,50]	28
(50,100]	15
(100,200]	6

Assume a uniform distribution of claim sizes within each interval.

Calculate [math]E(X^2) - E( (X \wedge 150)^2 ][/math]

• Less than 200
• At least 200, but less than 300
• At least 300, but less than 400
• At least 400, but less than 500
• At least 500

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

Key: C

In general,

[[math]]\begin{aligned} \operatorname{E}(X^{2})-E\left[(X \wedge 150)^{2}\right] &=\int_{0}^{200} x^{2} f(x) d x-\int_{0}^{150} x^{2} f(x) d x-150^{2} \int_{150}^{200} f(x) d x \\ & =\int_{150}^{200}\left(x^{2}-150^{2}\right) f(x) d x \end{aligned}[[/math]]

Assuming a uniform distribution, the density function over the interval from 100 to 200 is [math]6 / 7400[/math] (the probability of [math]6 / 74[/math] assigned to the interval divided by the width of the interval). The answer is

[[math]]\int_{150}^{200}\left(x^{2}-150^{2}\right) \frac{6}{7400} d x=\left.\left(\frac{x^{3}}{3}-150^{2} x\right) \frac{6}{7400}\right|_{150} ^{200}=337.84[[/math]]

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