The time to failure of a component in an electronic device has an exponential distribution with a median of four hours.

Calculate the probability that the component will work without failing for at least five hours.

• 0.07
• 0.29
• 0.38
• 0.42
• 0.57

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

Solution: D

The distribution function of an exponential random variable [math]T[/math] with parameter [math]\theta [/math] is given by [math]F(t) = 1 − e^{− t \theta} , t \gt 0 [/math]. Since we are told that [math]T[/math] has a median of four hours, we may determine [math]\theta[/math] as follows:

[[math]] \frac{1}{2} = F(4) = 1-e^{-4/\theta} \Rightarrow \theta = \frac{4}{\ln(2)}. [[/math]]

Therefore,

[[math]] \operatorname{P}(T ≥ 5 ) = 1 − F ( 5 ) = e^{−5 \theta} = e^{-\frac{5\ln(2)}{4}} = 2^{-5/4} = 0.42. [[/math]]

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