# 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

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?)