An insurance company sells two types of auto insurance policies: Basic and Deluxe. The time until the next Basic Policy claim is an exponential random variable with mean two days. The time until the next Deluxe Policy claim is an independent exponential random variable with mean three days. Calculate the probability that the next claim will be a Deluxe Policy claim.

• 0.172
• 0.223
• 0.400
• 0.487
• 0.500

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

Solution: C

Let [math]T_1[/math] be the time until the next Basic Policy claim, and let [math]T_2[/math] be the time until the next Deluxe policy claim. Then the joint pdf of [math]T_1[/math] and [math]T_2[/math] is

[[math]] f(t_1,t_2) = \left( \frac{1}{2} e^{-t_1/2}\right) \left( \frac{1}{3}e^{-t_2/3}\right) = \frac{1}{6}e^{-t_1/2}e^{-t_2/3}, 0 \lt t_1 \lt \infty, 0 \lt t_2 \lt \infty [[/math]]

and we need to find

[[math]] \begin{align*} P[T_1 \lt T_2] = \int_0^{\infty}\int_0^{t_1} \frac{1}{6} e^{-t_1/2}e^{-t_2/3} dt_2dt_1 &= \int_0^{\infty} \left[ -1\frac{1}{2} e^{-t_1/2}e^{-t_2/3}\right ]_0^{t_1} dt_1 \\ &= \int_0^{\infty}[\frac{1}{2}e^{-t_1/2} - \frac{1}{2}e^{-t_1/2}e^{-t_1/3}] dt_1 \\ &= \int_0^{\infty} \left[ \frac{1}{2}e^{-t_1/2} - \frac{1}{2} e^{-5t_1/6}\right] dt_1\\ & = \left [ -e^{-t_1/2} + \frac{3}{5}e^{-5t_1/6}\right]_0^{\infty} \\ &= 1- \frac{3}{5} = \frac{2}{5} \\ & = 0.4 \end{align*} [[/math]]

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