Poisson distribution, the meaning of the parameter lambda

Question

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

Answer 1

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.

Answer 2

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.