If $X$ conditional on $Y$ yields a certain distribution, how can we formally express $X|Y$ as a random variable?

Question

Suppose, $X,Y$ are random variables and $X$ conditional on $Y$ yields a certain distribution, how can we formally express $X|Y$ as a random variable? I know that writing $X|Y$ is abusive in notation but what is the proper way of expressing this? I surmise that we need to use indicator random variables, does anyone know?

Answer 1

Indeed $X\mid Y$ is not an object, in itself, but writing $X\mid Y\sim\mathcal{Distribution}$ is acceptable notation; it says that $X$ has that distribution when conditioned by $Y$.

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If $X$, when given $Y$, has a conditional uniform distribution over $[0;Y]$ we would write:$$X\mid Y~\sim~\mathcal U[0;Y]$$ Which tells us that: $$f_{X\mid Y}(x\mid y)=y^{-1}\cdot\mathbf 1_{x\in[0;y]}$$

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It is also acceptable to write $~X\mid Y{=}y ~\sim~\mathcal {Bin}(y, p)~$, which in this case means that $X$ is binomially distributed, given $Y=y$, with parameters: trials of $y$, and success rate of $p$.

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And such.