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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, $Q(\{x : x \in \mathcal X, f(x) = y\})$.

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

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 \} $.


[1] According to this metric, I am rather witless. :\

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  • $\begingroup$ Thanks for the thorough answer! I agree with you there. $\endgroup$
    – AlCorreia
    Mar 2 '20 at 8:01

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