Two identical components having lifetimes $A$ and $B$ are connected in parallel in a system
.Suppose the distributions of $A$ and $B$ independently follow exponential with mean $\frac 1a, a>0$. But whenever one component fails the lifetime distribution of another changes to exponential with mean $\frac 1c ,c>0$ .Let $T$ denote the overall lifetime of the system
Find $P(T\ge t),\:t>0$
I am understanding that the total time will be $\min (A,B)+ \exp(c)$ but not getting how to write integral and also the limits of integration please help!!
Best Answer
The time until the first failure is the minimum of two exponential random variables; one can show that this is itself an exponential random variable with mean $\frac{1}{2a}$.
The time between the first failure and the second failure is, by assumption, exponential with mean $1/c$.
Thus $T$ is the sum of two independent exponential random variables with different means. You can use the convolution formula to get the PDF of $T$, e.g. see here.