# What is the probability the lifetime of two electric bulbs with exponential distribution exceeds 2λ?

Let $X$ be the lifetime of a certain electric bulb, and $Y$ the lifetime of its replacement after the failure of the first bulb. Suppose $X$ and $Y$ are independent with common exponential density function with parameter $\lambda$.

(a) Find the probability that the combined lifetime exceeds $2\lambda$.

(b) What is the probability that the replacement outlasts the original component by $\lambda$?

Any help is appreciated.

• a) Find the distribution of $X+Y$; b) Find the distribution of $Y-X$. – André Nicolas May 12 '16 at 0:33

## 1 Answer

a) $\mathsf P(X+Y\geq 2\lambda)$.   If $X,Y$ are independent exponential distributed random variables, do you know the distribution of their sum? (Or how to find it?)

b) $\mathsf P(Y-X\geq \lambda)$.   Likewise, do you know the distribution of their difference? (Or how to find it?)

Are you expected to work from first principles or do you have previously know results available in your notes?

• Got it, thanks. I solved it (at least I think) but don't know how to type it here – user338895 May 14 '16 at 1:13