# Poisson distribution, the meaning of the parameter lambda

What is the best intuition behind the unique parameter $$\lambda$$ in the Poisson distribution?

Poisson RV is commonly used for modelling number of occurrences of an event within a particular time interval. And, since $$E[X]=\lambda$$, its unique parameter is referred as mean number of event occurrences within our particular time interval.

• That's nice. Can be something be added about the "particular time inetrval" ? – user122424 Oct 21 '18 at 12:29
• e.g. $X$: number of customers arriving daily; here time interval is day. – gunes Oct 21 '18 at 12:31
• Is the Poisson distribution strictly speaking discrete or can it be even continuous ? I.e. can we say that there is a probability of number customers arrving daily 2.45 ? – user122424 Oct 21 '18 at 12:42
• Poisson is a discrete RV. It can be 0,1,2,... – gunes Oct 21 '18 at 12:48
• Yes but it has continuous analog if we intersperse curve through the discrete values at $0,1,2,...$ Can this be done and understood formally? – user122424 Oct 21 '18 at 13:00

Since the Poisson distribution is defined as

$$\pi_{\lambda}(k)~=~\frac{\lambda^k}{k!}e^{-\lambda}$$

It is not hard to show that $$E[X]=\lambda$$ and $$D^2[X]=\lambda$$. Therefore you can directly interpret $$\lambda$$ as the expectation or as well as the variance of the poisson distributed variable.