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

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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.

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  • $\begingroup$ That's nice. Can be something be added about the "particular time inetrval" ? $\endgroup$ – user122424 Oct 21 '18 at 12:29
  • $\begingroup$ e.g. $X$: number of customers arriving daily; here time interval is day. $\endgroup$ – gunes Oct 21 '18 at 12:31
  • $\begingroup$ 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 ? $\endgroup$ – user122424 Oct 21 '18 at 12:42
  • $\begingroup$ Poisson is a discrete RV. It can be 0,1,2,... $\endgroup$ – gunes Oct 21 '18 at 12:48
  • $\begingroup$ 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? $\endgroup$ – user122424 Oct 21 '18 at 13:00
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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.

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