# Counting random exponential events

Let's say we have random events happening with the exponential probability: $$P(t_i>T)=e^{-\lambda T}$$, where $$i=1,..,n$$. What is the probability that there will be $$k$$ events within time period $$(0,T)$$? My understanding is that it should be a Poisson distribution with mean $$\lambda$$, but why and how can I prove it rigorously?

• Do you mean that the time between events is exponential? The way you have said it right now, it sounds like the time that the $i^{th}$ event occurs is exponentially distributed, which would mean the number of events which occur in $(0,T)$ would have the Binomial distribution $\text{Bin}(n,1-e^{-\lambda T})$, not Poisson. (This is also assuming the times are independent, which you have not specified). Can you clarify? Mar 12, 2020 at 14:44
• You are right. Can you write it as an answer? I will accept it. I think the book I am reading is just wrong in claiming it is a Possion distribution. It would become a Poisson if probability is low. Mar 15, 2020 at 22:25