An insurance company sells automobile liability and collision insurance. Let [math]X[/math] denote the percentage of liability policies that will be renewed at the end of their terms and [math]Y[/math] the percentage of collision policies that will be renewed at the end of their terms. [math]X[/math] and [math]Y[/math] have the joint cumulative distribution function

[[math]] F (x,y) = \frac{xy(x+y)}{2,000,000}, \, 0 ≤ x ≤ 100, \, 0 ≤ y ≤ 100. [[/math]]

Calculate [math]\operatorname{Var}(X).[/math]

• 764
• 833
• 3402
• 4108
• 4167

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

Solution: C

The probability needs to be calculated for each total number of claims. 0: 0.5(0.2) = 0.10 1: 0.5(0.3) + 0.3(0.2) = 0.21 2: 0.5(0.4) + 0.3(0.3) + 0.2(0.2) = 0.33 3: 0.5(0.1) + 0.3(0.4) + 0.2(0.3) + 0.0(0.2) = 0.23 At this point there is only 0.13 probability remaining, so the mode must be at 2.

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