The lifetime of a certain electronic device has an exponential distribution with mean 0.50. Calculate the probability that the lifetime of the device is greater than 0.70, given that it is greater than 0.40.

• 0.203
• 0.247
• 0.449
• 0.549
• 0.861

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

Solution: D

The distribution function of an exponential distribution with mean 0.5 is [math]F(x) = 1-e^{-2x}[/math].

[[math]] \operatorname{P}(X \gt 0.7 | X \gt 0.4 ) = \frac{\operatorname{P}(X \gt 0.7)}{\operatorname{P}(X \gt 0.4)} = \frac{e^{-1.4}}{e^{-0.8}} = 0.549. [[/math]]

This can be more efficiently solved using the memoryless property:

[[math]] \operatorname{P}( X \gt 0.7 | X \gt 0.4 ) = \operatorname{P}(X \gt 0.3) = e^{-0.6} = 0.549. [[/math]]

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