Why is F used for an unspecified distribution?

Question

I'm taking a linear regression course in university right now, and my professor often uses the notation F to refer to a generic, unspecified distribution with the given mean and variance. For example, $\epsilon_i \stackrel{uncorr.}{\sim} F(0, \sigma^2)$. Sometimes he'll even just exclude the F, saying $\epsilon_i \stackrel{uncorr.}{\sim} (0, \sigma^2)$. I'm just curious why this notation is used, especially considering we use F-tests often, and it's very confusing for me to know when he's referring to one or the other.

Answer 1

There is hardly a letter you would want to use in the usual Western and Greek alphabets that doesn't have multiple frequent uses in statistics. Reversing the question, are there more than about 100 characters in statistical plays? (I get 100 or so from two alphabets, upper and lower cases, and some letters usually avoided as unfamiliar ($\Xi$?) Yes. Do people disagree on their favourite or customary notation? Yes. Hence some clashes of notation are inevitable.

$a$ is often an intercept, a frequency in a $2\times2$ table, ...

$b$ is often a gradient, another such frequency, ...

and $F$ is often used for a generic or specific (cumulative) distribution function, which may be what suggested this notation. As you say, it's also often used in honour of R.A. Fisher as a statistic, a ratio of variances or mean squares in analysis of variance, regression and so forth, and also for the associated $F$ distribution.

The most important points about using notation are

1. Every teacher or author has to assume some prior knowledge, as no one wants to recapitulate your entire mathematical education. Where one draws the line is a matter of assumed prerequisites and of context, style and taste. Some authors feel obliged to explain $e = \exp(1) \approx 2.71828$ while others will assume much more, that you recognise $\Gamma()$ as a gamma function, that you know conventional symbols for widely used spaces and groups, and so forth.

2. But teachers and authors do have obligations to explain their notation, to be consistent at least locally, and (this is a little arguable) to use notation only when it helps. (Using a symbol just once can often be avoided.)

3. It's also often true that readers pick up notation by example rather than by precept, just as children don't mostly learn new words by looking in dictionaries. A good notation often suggests its own explanation, as (e.g.) $1(1)100$ and $1(1)10(10)100$ might well do.

P.S. Why not ask the Professor?