Let [math]X[/math] be a random variable with density function

[[math]] f(x) = \begin{cases} 2e^{-2x}, \, x \gt0 \\ 0, \, \textrm{otherwise} \end{cases} [[/math]]

Calculate [math]\operatorname{P}[ X \leq 0.5 | X \leq 1.0].[/math]

• 0.433
• 0.547
• 0.632
• 0.731
• 0.865

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

Solution: D

[[math]] \begin{align*} F(x) &= \int_0^{x}2e^{-2y} dy = -e^{-2y} \Big |_0^x = 1-e^{-2x} \\ P[X \leq 0.5 | X \leq 1.0] &= \frac{P[X \leq 0.5]}{P[X \leq 1.0]} = \frac{F(0.5)}{F(1.0)} = \frac{1-e^{-1}}{1-e^{-2}} = 0.731. \end{align*} [[/math]]

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