An insurer offers a travelers insurance policy. Losses under the policy are uniformly distributed on the interval [0, 5]. The insurer reimburses a policyholder for a loss up to a maximum of 4.

Determine the cumulative distribution function, [math]F[/math], of the benefit that the insurer pays a policyholder who experiences exactly one loss under the policy.

• [[math]]F(x) = \begin{cases} 0, \, x \lt 0 \\ 0.20x, \, 0 \leq x \lt 4 \\ 1, \, x \geq 4 \end{cases}[[/math]]
• [[math]]F(x) = \begin{cases} 0, \, x \lt 0 \\ 0.20x, \, 0 \leq x \lt 5 \\ 1, \, x \geq 5 \end{cases}[[/math]]
• [[math]]F(x) = \begin{cases} 0, \, x \lt 0 \\ 0.25x, \, 0 \leq x \lt 4 \\ 1, \, x \geq 4 \end{cases}[[/math]]
• [[math]]F(x) = \begin{cases} 0, \, x \lt 0 \\ 0.25x, \, 0 \leq x \lt 5 \\ 1, \, x \geq 5 \end{cases}[[/math]]
• [[math]] F(x) = \begin{cases} 0, \, x \lt 0 \\ 0.25x, \, 1 \leq x \lt 5 \\ 1, \, x \geq 5 \end{cases} [[/math]]

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

Solution: A

Let Y be the loss and X be the reimbursement. If the loss is less than 4,P( X ≤ x)= P (Y ≤ x)= 0.2 x for x < 4 because Y has a uniform distribution on [0, 5]. However, the probability of the reimbursement being less than or equal to 4 is 1 because 4 is the maximum reimbursement.

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