johndcook.com
Distinguishing variables from parameters
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.
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.
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.
