I know how to calculate the probability of one event taking place within a set time period with exponential distribution but I'm having difficulty figuring out how to calculate what would happen if there was a second event that took place within a certain time
For example how would I calculate the probability of two people arriving within 6 minutes when the mean is 3 minutes using exponential distribution.
Best Answer
The exponential has parameter $\lambda=\frac{1}{3}$.
The number of "events" in a $t$ minute interval has Poisson distribution parameter $\frac{t}{3}$.
Thus the probability of exactly $2$ events in a time interval of length $t$ is $$e^{-t/3}\frac{(t/3)^2}{2!}.$$