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I'm in a setting in which X and Y are both $Beta(0.5, 0.5)$ and indipendent, that is

$$f(x, y) = \frac{1}{\pi^2\sqrt{x(1-x)y(1-y)}}$$

What is the conditional distribution of X given Y = X? Normally i would substitute x to y in the density function and i would find a normalization constant. But in this case we would have

$$f(x) \propto \frac{1}{x(1-x)}$$

which is not integrable. But it feels strange to me that a well defined distribution on a bidimensional set, becomes undefined when restricted to a subspace of the original domain. Am I doing something wrong? There actually exist the pdf of this conditional distribution?

Ivan
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    The usual thing is to ask about the distribution of $X$ given $g(x,y)=a$, whose cdf is defined as $$F(x) = \lim_{\epsilon\to 0^+}P[X<x:|g(x,y)-a|<\epsilon]$$ But beware that conditioning on $x-y=0$ may give different results from conditioning on $x^2-y^2=0$. – Matt F. Mar 24 '24 at 01:12
  • Hint (for intuition and a solution): re-express the variables as $x=\sin\theta$ and $y=\sin\phi.$ The apparent problem will disappear provided you do the calculation correctly. After verifying that, see whether you can translate this result into the original variables. – whuber Mar 24 '24 at 15:53

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