# Notation: probability of realizations satisfying a given condition

Let $$X$$ be a random variable distributed according to $$Q$$ and $$f$$ an arbitrary function defined on the sample space of $$X$$, $$f :\mathcal X \rightarrow \mathbb R$$.

I often see the following notation used for the likelihood (under $$Q$$) of realizations of $$X$$ for which $$f$$ assumes a given value $$y$$.

$$Q(f(X) = y)$$

Is this notation standard or even appropriate? Is there a better alternative for a concise notation?

I find it confusing as it is not the probability of the event $$f(X) = y$$ itself, but rather the probability of realizations of $$X$$ that satisfy this condition, that is, $$Q(\{x : x \in \mathcal X, f(x) = y\})$$.

• This is a standard abuse of notation universally accepted by probabilists. Feb 29 '20 at 11:44
• Thanks, @KaviRamaMurthy! That's all I wanted to know. Feb 29 '20 at 11:45

Expanding on this a little, remember that the point of notation is to clearly communicate ideas, and that brevity is the soul of wit.[1] One could write $$Q(\{x : x\in\mathcal{X}, f(x) = y\}),$$ but this notation is rather bulky, is almost never going to read well inline, and contains a lot of redundancy. It is much simpler to write $$Q(f(X)=y)$$ and, as this can be done without sacrificing clarity, it is an entirely appropriate notation.
It may also be worth noting that this is a kind of simplified notation which shows up in other domains. For example, in complex analysis / analytic number theory, sets of complex numbers are often specified in a similar manner, e.g. $$\{\Re(s) > 1\}$$ is the open half-plane $$\{s \in \mathbb{C} : \Re(s) > 1 \}$$.