5Why do they use "the" in the first sentence? It is first mentioning and it should be without article (if it is uncountable noun) and with a/an if it is countable.

Asappears to be an alternative name forthe Poisson Distribution, the definite article is fine.the Poisson law of small numbers

[...]would it be OK to write it without "the" completely, such as

"In probability theory and statistics, Poisson distribution ...is a discrete probability distribution that expresses the probability of a"?

If you mean "distribution in the way noted by Poisson", then probably yes.

Academic

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