# 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: $$P(\cdot)$$

• $\cdot$ is sometimes used as a placeholder, signifying "This is where you would put something". For instance, I can show you how we in modern mathematics usually write square roots by writing $\sqrt{{}\cdot{}}$. That might be what's happening here, but without more context is hard to say. – Arthur Sep 26 '16 at 16:51
• Exactly what @Arthur wrote, except $P(\cdot)$ or $\mathbb{P}[\cdot]$ usually stand for the probability of some event – gt6989b Sep 26 '16 at 16:55
• So they may write $P(\cdot)$ when they want to refer to the whole system of values $P(E)$, with $E$ ranging over all possible events. – GEdgar Sep 26 '16 at 17:06
• Yes I think a placeholder makes sense, thanks. – Amour Sep 26 '16 at 17:09

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).