Poisson distribution concrete example

Question

What is an example of a Poisson distributed random variable with parameter $\lambda > 0$?

Answer 1

From Wikipedia's entry on the Poisson distribution:

For instance, a call center receives an average of 180 calls per hour, 24 hours a day. The calls are independent; receiving one does not change the probability of when the next one will arrive. The number of calls received during any minute has a Poisson probability distribution with mean 3: the most likely numbers are 2 and 3 but 1 and 4 are also likely and there is a small probability of it being as low as zero and a very small probability it could be 10. Another example is the number of decay events that occur from a radioactive source during a defined observation period.

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UPDATE

I don't quite understand what a more mathematical example would be. Perhaps if you have a discrete random variables $X,Y,Z$ with the pmfs $$ \begin{split} f_X(k) &= \frac{1/e}{k!}, \quad k \ge 0 \\ g_Y(k) &= \frac{e^{k-e}}{k!}, \quad k \ge 0 \\ h_Z(k) &= \frac{1}{e^\pi} \frac{\pi^k}{k!}, \quad k \ge 0 \end{split} $$ then $X \sim \mathcal{P}(1), Y \sim \mathcal{P}(e)$ and $Z \sim \mathcal{P}(\pi)$.