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$X$ and $Y$ are random variables. $X$ has uniform distribution in $[-1,1]$, i.e, $F_{X} = 1/2$ in $[-1,1]$, and 0c.c. $Y = X^{2}$

What are the distribution of $f_{Y|X=x}$ and $f_{X|Y=y}$?

I found that the marginal o Y is: $f_{Y} = 1/2\sqrt{y}$. Am I right?

And then I, just stuck here: $f_{Y|X=x} = f_{x,y}/f_{x} = 2 f_{x,y}$

What can I do after this?

Any help?

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  • $\begingroup$ If $Y=X^2$, none of the PDFs $f_{X,Y}$, $f_{Y\mid X=x}$ and $f_{X\mid Y=y}$ exist. But, conditionally on $X=x$, the distribution of $Y$ is a Dirac mass at $x^2$, and, conditionally on $Y=y$, $X$ is uniformly distributed on $\{-\sqrt y,\sqrt y\}$. Is this what you are asking? $\endgroup$ – Did Jan 19 at 17:53
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    $\begingroup$ @Did Exactly! But I will need the $f(x,y)$ distribution, right? How can I find her? $\endgroup$ – Laura Jan 19 at 17:58
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    $\begingroup$ Hmmm... Did you read my comment? Apparently not, so let me repeat: the PDF $f_{X,Y}$ does not exist. $\endgroup$ – Did Jan 19 at 18:06
  • $\begingroup$ Laura you may want to start with Dirac Delta. $\endgroup$ – Lee David Chung Lin Jan 20 at 10:42
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    $\begingroup$ @Mitjackson thank you very much! Now I get it. In some basic probability books this type of example are not found. Could you suggest a book, or any pdf that I could find examples like that? $\endgroup$ – Laura Feb 28 at 15:04

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