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I am slightly confused about the use of indicator variables in probability density functions.

For example, consider the density of $(X,Y)$ uniformly distributed on the unit disc. This density can be written as $$f_{X,Y}(x,y)= \frac{1}{\pi} \mathbb{1} \left \{ (x,y) \mid x^2 + y^2 \leq 1 \right \}.$$

However, I am not really seeing how this makes sense. I understand indicator variables when solving problems with probability, but why not just write the density without it? What is it exactly saying?

Thanks in advance.

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Note that $f_{X,Y}:\mathbb R^2\to\mathbb R$.

It says exactly that $f_{X,Y}(x,y)$ will take value $\frac1{\pi}$ if $x^2+y^2\leq1$ and will take value $0$ otherwise.

So an excellent presentation of a PDF which is formally for a fixed nonnegative integer $n$ a nonnegative measurable function $\mathbb R^n\to\mathbb R$ that gives value $1$ by integration wrt Lebesgue-measure.

In notation it can be handsome to write things like: $\mathbb EX=\int f_X(x)xdx$ without bothering on borders.


If it is not your taste then of course you can also choose for discerning cases: $x^2+y^2\leq1$ and "otherwise" without any mentioning of indicatorfunction.

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