Automobile claim amounts are modeled by a uniform distribution on the interval [0, 10,000]. Actuary A reports [math]X[/math], the claim amount divided by 1000. Actuary B reports [math]Y[/math], which is [math]X[/math] rounded to the nearest integer from 0 to 10.

Calculate the absolute value of the difference between the 4^th moment of [math]X[/math] and the 4^th moment of [math]Y[/math].

• 0
• 33
• 296
• 303
• 533

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

Solution: B

The fourth moment of [math]X[/math] is

[[math]] \int_0^{10} \frac{x^4}{10} dx = \frac{x^5}{50} \Big |_0^{10} = 2000. [[/math]]

The [math]Y[/math] probabilities are 1/20 for [math]Y = 0 [/math] and 10, and 1/10 for [math]Y = 1,2, \ldots, 9 [/math].

[[math]] \operatorname{E}[Y^4] = (1^4 + 2^4 + \cdots + 9^4)/10 + 10^4/20 = 2033.3. [[/math]]

The absolute value of the difference is 33.3.

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