# Why are the geometric distribution and hypergeometric distribution called as such?

Why are the geometric distribution and hypergeometric distribution called “geometric” and “hypergoemetric” respectively?

Is it because their pmfs take some special form? Thanks!

Yes, the terms refer to the probability mass functions (pmfs).

2,500 years ago, Euclid (in Books VIII and IV of his Elements) studied sequences of lengths having common proportions.. At some point such sequences came to be known as “geometric progressions” (although the term “geometric” could for a similar reason just as easily have been applied to many other regular series, including those now called “arithmetic”).

The probability mass function of a geometric distribution with parameter $p$ forms a geometric progression

Here the common proportion is $1-p$.

Several hundred years ago a vast generalization of such progressions became important in the studies of elliptic curves, differential equations, and many other deeply interconnected areas of mathematics. The generalization supposes that the relative proportions among successive terms at positions $k$ and $k+1$ could vary, but it limits the nature of that variation: the proportions must be a given rational function of $k$. Because these go “over” or “beyond” the geometric progression (for which the rational function is constant), they were termed hypergeometric from the ancient Greek prefix $\grave\upsilon^\prime\pi\varepsilon\rho$ (“hyper”).

The probability mass function of a hypergeometric function with parameters $N, K,$ and $n$ has the form

for suitable $k$. The ratio of successive probabilities therefore equals

a rational function of $k$ of degree $(2,2)$. This places the probabilities into a (particular kind of) hypergeometric progression.