The random variable [math]X[/math] represents the random loss, before any deductible is applied, covered by an insurance policy. The probability density function of [math]X[/math] is

[[math]] f(x) = 2x, \, 0 \lt x \lt 1. [[/math]]

Payments are made subject to a deductible, [math]d[/math], where 0 < [math]d[/math] < 1.

The probability that a claim payment is less than 0.5 is equal to 0.64.

Calculate the value of [math]d[/math].

• 0.1
• 0.2
• 0.3
• 0.4
• 0.5

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

Key: C

[[math]] F(x) = \int_0^{x} 2ydy = x^2. [[/math]]

Let [math]C[/math] be a random claim payment. Then [math]C = 0 [/math] if [math]X \lt d [/math] and [math]C = X-d [/math] if [math]X \geq d [/math]. Then,

[[math]] \begin{aligned} \operatorname{P}(C \lt 0.5) = 0.64 \\ \operatorname{P}(C \geq 0.5) = 0.64 \\ \operatorname{P}( X − d \geq 0.5) = 0.36 \\ \operatorname{P}( X \geq 0.5 + d ) = 0.36 \\ F (0.5 + d ) = 0.64 \\ (0.5 + d )^ 2 = 0.64 \\ 0.5 + d = 0.8 \\ d = 0.3 \\ \end{aligned} [[/math]]

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