Question

A light bulb has a lifetime X that is exponentially distributed with a mean 340 hours. Find the probability that the bulb lifetime exceeds 220 hours when you know it already exceeded 100 hours ?

Answer 1

0.7026

Explanation:

Let X denote the lifetime of light bulb. Given
`X \sim Exp(\lambda)` where the mean is
`E(X) = 340 =(1)/(\lambda) \implies \lambda = (1)/(340) = 0.00294`.

Recall that,

`\displaystyle P(X>x) = 1 - \int_0^x e^(-\lambda x) = 1 - (1 - e^(-\lambda x) = e^(-\lambda x)`

`\displaystyle P(X>220 | X>100) = (P(X>220,X>100))/(P(X>100)) = (e^(-\lambda * 220))/(e^(-\lambda * 100)) = (0.5236)/(0.7452) = 0.7026`