Plausible to assume that a certain random variable has a Poisson distribution > Why it might be plausible to assume that the following random variables follow a Poisson distribution? : $a)$ The number of customers that enter a store in a fixed time period. $b)$ The number of customers that enter a store and buy something in a fixed time period. $c)$ The number of atomic particles in a radioactive mass that disintegrate in a fixed time period. $ a)$ My Solution: I think in this case is plausible to assume that the random variable follow a Poisson distribution because of the Law of rare events, i.e. we are considering a large number $n$ of independent events, each of which has a small success probability $\lambda/n$. The number $n$ in this case is the number of customers that could enter the store, clearly independent. But small probability for each to actually enter. Is my argument correct? Am I missing something?

Your argument is correct. A purist might say the Poison distribution involved the limit as $n\to\infty$ and so it's merely an approximation when $n$ is finite. But that degree of purity is actually a bit out of place in practical mathematical modeling.