# What is mean by the term “constant rate” in Poisson distribution?

I have difficulty in understanding the assumptions of Poisson distribution, one assumption is the rate at which the events occur in the time interval is constant. What is the meaning of that phrase? I would highly appreciate if you give me an intuitive example (not a rigorous mathematical one).

Thanks

• You are confusing the "Poisson distribution" with the "Poisson process".They are not the same. The Poisson distribution has one parameter (which of course is constant). The Poisson process has a rate which can be constant or not. – Zahava Kor Jul 25 '18 at 14:13

The typical intuitive example is this: suppose we are interested in finding the probability distribution of the number of cars that will drive through a particular intersection during a period of one hour. One strategy would be to divide this one-hour period into many, very small periods: periods that are so small, that the probability that more than one car will cross the intersection during each period is negligible (i.e. zero). These $n$ very small periods can now be thought of as containing Bernoulli random variables $x$, with $p$ being the probability of observing a car, and $(1-p)$ the probability of not observing a car. The sum of these Bernoulli random variables would then give us the number of cars for an hour and, assuming they are i.i.d., this sum is Binomial distributed. But, how can we choose an appropriate $n$? With $n=\infty$, that Binomial distribution actually becomes the Poisson distribution.
Finally, here is the intuition you are looking for. If the number of cars you observe each hour (the number of events in each time period $t$) is Poisson distributed, then the time that passes between each event of observing a car has an Exponential distribution. A process like this, where the counts are Poisson and the durations are Exponential, has a constant hazard rate.