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Why 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.
Given that it is uncounbale possibility, 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"
OK so I guess it comes from the fact that the word law is countable, and the term referrs to one such law. There is a one to one correspondence.
Well, I think I start to understand. If a distribution is related to some physical phenomena that is well known, then "the" is used.
Gee, this starts making sense... Thanks!
My understanding is that when we use "the", we don't want to see many Poisson distributions. We want to see them all as one general notion. It may not be right formally, since a distibution is a concrete measure (or a concrete something else, depending on what your favourite definition is), not a class of measures. Poisson distributions form a class of probability distributions indexed by positive real numbers lambda. We sometimes call this class the Poisson distribution perhaps because "the class of Poisson distributions" is cumbersome and doesn't give us anything other than formal correctness.As the Poisson Distribution appears to be an alternative name for the Poisson law of small numbers, the definite article is fine.
This is only a non-native's opinion. I would recommend that you do indeed post on a mathematics-related forum. A native mathematician's opinion will be worth more than anything you can get here, (Unless we do have a native mathematician on these forums -- I do not know that.)