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.