Imagine the following dialog.

Professor: f is a function of a real variable x that takes a real parameter k.

Student: What’s a parameter?

Professor: It’s a constant that can vary.

Student: Then if it can vary, isn’t it a variable?

Professor: Sorta, but no not really.

This conversation plays out over and over, and unfortunately it often ends as it does above, with the student confused. Here’s how I believe the conversation should continue.

Professor: You’re absolutely right that f is a function of two variables, x and k. But usually k is fixed in the context of a specific application and x is not. A different application might have a different, but also fixed, value of k. So it is helpful to think of f(x; k), a function of x with a parameter k, rather than f(x, k), a function of two variables. The former carries more information, giving a hint as to how the numbers are used.

Is there really a difference between a parameter and a variable? In a reductionistic sense, no. But in a practical sense, yes, absolutely.

It might sound pedantic to distinguish a variable from a parameter, and it is, in the best sense of the word. Pedant literally means teacher. Usually pedantic carries a negative connotation, such as making a distinction without a difference. But here the pedant would be making a helpful distinction.

For example, we might write a probability density function as f(x; μ, σ). The function gives the probability density at a point x. The density depends on parameters μ and σ, and these parameters change between applications, but for a given application they have fixed values.

You find the probability of a random variable taking on values in an interval [a, b] by integrating f over that interval. When I say that, you know that I mean you’d integrate with respect to x, because f is a function of x. It is also, in an abstract sense, a function of μ and σ, but it’s typically not useful to think of it that way.

Hypergeometric functions have two sets of parameters, and so you may see two semicolons, such as f(x; a, b; c). This denotes a function of the variable x, with upper parameters a and b, and a lower parameter c. In some abstract sense this is a function of four variables, but it acts very differently with respect to x than with respect to a, b, and c. There’s also a difference between a and b on the one hand an c on the other, one worth paying attention to, though it is less of a difference than between x and the parameters collectively.

Sometimes you’ll see a vertical bar rather than a semicolon to separate variables from parameters. This works out even better for probability densities because then f(x | μ, σ) suggests the probability density of x given μ and σ since the vertical bar is also used for conditional probability. You might also see f(x | a, b; c) for hypergeometric functions, with the vertical bar separating variables from parameters and the semicolon separating two kinds of parameters.

When I first saw a semicolon separating variables from parameters, no explanation was given, and I figured I could mentally replace the semicolon with a comma. Then later I realized that the semicolon was an act of kindness by the author giving the reader additional information.