# Finding the distribution of $f_{Y|X=x}$ and $f_{X|Y=y}$

$$X$$ 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?

• 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? – Did Jan 19 at 17:53
• @Did Exactly! But I will need the $f(x,y)$ distribution, right? How can I find her? – Laura Jan 19 at 17:58
• Hmmm... Did you read my comment? Apparently not, so let me repeat: the PDF $f_{X,Y}$ does not exist. – Did Jan 19 at 18:06
• Laura you may want to start with Dirac Delta. – Lee David Chung Lin Jan 20 at 10:42
• @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? – Laura Feb 28 at 15:04