What does the centre dot notation mean? $P(\cdot)$

I think the notation has something to do with probability. I saw the symbol somewhere and I cannot seem to find anything on it. If I were to read what it looks like, I would say: probability as a function of dot.

It looks like this: $P(\cdot)$

Thanks in advance!

Solution 1:

Remember that a function $f:A \to \Bbb R$ is an object defined such that, if we select an element $a$ of $A$, $f(a)$ is a number.

When one is introduced to functions, it is common to refer to $f$ itself as the function "$f(x)$". In this context, $x$ should be interpreted as a "dummy variable". In this case, that means that if we want to evaluate $f$ at an element $a$, we should replace $x$ with $a$. For example, if $f: \Bbb R \to \Bbb R $ is the function $f(x) = x^2 + 2$, then to find $f(1)$ we replace $x$ with $1$ to compute $f(1) = (1)^2 + 2 = 3$. The letter $x$ had nothing to do with the function itself; $f(y) = y^2 + 2$ defines precisely the same function.

This idea of a "dummy variable" can be ambiguous and therefore confusing. In particular, $x$ could refer to a specific element of $A$. If that's the case, then $f(x)$ is a specific number as opposed to a function. In the above example, if we were to say that $x = 1$, then $f(x)$ would refer to the number $3$, rather than the function $f$ itself.

With that in mind: in order to distinguish between "$f(x)$ the function" and "$f(x)$ the number", it is common to use $f(\cdot)$ to refer specifically to the function (as opposed to any particular output value).