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There are some probability distributions named after Greek letters, such as Gamma, Beta, Chi, and Zeta distributions.

I wonder if there are some rules of why and how they are named after Greek letters?

Can we make use of the rules (if exist) to help us memorize these distributions?

Are there other such distributions?

Thanks and regards!

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I think that there are no such rules; there is only history.

Whether to generalize rules that don't exist is a very good question. I think sometimes that happens. In other words, no rule brought this about, but maybe by coincidence they fit some simple pattern that no one's noticed, and once we do notice, we can extrapolate, and that extrapolation may be valuable. But simplicity is essential here. Obviously there's such a rule if we don't require simplicity, namely: the naming conventions are precisely that we call this one the Beta distribution, and this one the Gamma distribution, etc. And it's easy to extrapolate then!

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There are also conventional uses of lower-case Greek letters as parameters in parametrized families of probability distributions: $\mu$ is the population mean (and $\bar x$ is the sample mean), $\sigma^2$ is the population variance (and $S^2$ is the sample variance), $\rho$ is the population correlation (and $r$ or $R$ is the sample correlation), $\alpha$ and $\beta$ are the parameters for the Beta distribution, $\varepsilon$ is the (unobservable) error (and $\hat\varepsilon$ is the (observable) residual), $\theta$ is the general-purpose parameter used when you say "Consider a parametrized.... – Michael Hardy May 30 '13 at 23:38 of probability distributions", $\lambda$ is the most frequently used symbol for the mean of a Poisson distribution, $\pi$ is sometimes a population proportion (in $[0,1]$), (lower-case) $\varphi$ is the standard normal density (the "bell-shaped curve"), (capital) $\Phi$ is the corresponding c.d.f. – Michael Hardy May 30 '13 at 23:40
Thanks, Michael! I found that except chi distribution, the pdf of the other three are based on gamma, beta, and zeta functions. That is probably why they are named as such. I am not sure about chi distribution though. – Tim May 31 '13 at 23:04

There's a general problem with mathematics that every researcher eventually encounters. There aren't enough letters in the alphabet.

In probability (the only discipline I research) if you discover a new distribution then most of the time one of two things has happened.

1.$\qquad$You've proposed an exotic or special formula at the beginning of your paper so you give it an exotic name, like a Greek letter.

2.$\qquad$You've spent ten pages deriving the formula for your distribution and you've run out of normal letters.

So some of the time the new distributions get given Greek letters. Not always true, the $T$ distribution is important, $W$ is usually used for Brownian motion which is more important than all the other distributions put together.

You might find a pattern that helps to remember them, but it will be a coincidence, not a rule.

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