An insurance policy pays for a random loss [math]X[/math] subject to a deductible of [math]C[/math], where [math]0 \lt C \lt 1[/math] . The loss amount is modeled as a continuous random variable with density function

[[math]] f(x) = \begin{cases} 2x, \, 0 \lt x \lt 1 \\ 0, \, \textrm{otherwise.} \end{cases} [[/math]]

Given a random loss [math]X[/math], the probability that the insurance payment is less than 0.5 is equal to 0.64. Calculate [math]C[/math].

• 0.1
• 0.3
• 0.4
• 0.6
• 0.8

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

Solution: B

Denote the insurance payment by the random variable [math]Y[/math]. Then

[[math]] Y = \begin{cases} 0, \quad 0 \lt X \leq C \\ X-C, \quad C \lt X \lt 1 \end{cases} [[/math]]

Now we are given that

[[math]] \begin{align*} 0.64 = \operatorname{P}[Y \lt 0.5 ) = \operatorname{P}[ 0 \lt X \lt 0.5 + C ) &= \int_0^{0.5 + C} 2x dx \\ &= x^2 \Big |_0^{0.5 + C} \\ &= (0.5 + C)^2. \end{align*} [[/math]]

Therefore, solving for [math]C[/math], we find [math]C = ±0.8 − 0.5[/math]. Finally, since [math]0 \lt C \lt 1 [/math] , we conclude that [math]C = 0.3[/math].

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