Solved – Distribution of total number of car accidents in a lifetime of the car

Question

the number of car accidents every year has Poisson distribution with mean $\lambda$ and lifetime of the car has exponential distribution with mean $\frac{1}{\mu}$.
how can find distribution total number of car accidents in a lifetime of the car?

Answer 1

The Poisson distirbution with rate $\lambda t$ gives the distribution of the number of events in a time interval of length $t$. So the conditional distribution for number of events in the lifetime of the car is Poisson with rate $\lambda t$ given the car's life is $t$. Multiply this by the exponetial density for the lifetime of $t$ and integrate over all $t$ to get the distribution.