Mixing
Finding the distribution of $f_{Y|X=x}$ and $f_{X|Y=y}$
2
$begingroup$
$X$ 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: $f_{Y} = 1/2sqrt{y}$. 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?
probability probability-theory probability-distributions
asked Jan 19 at 16:26
LauraLaura
3018
$endgroup$
|
show 2 more comments
2
$begingroup$
$X$ 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: $f_{Y} = 1/2sqrt{y}$. 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?
probability probability-theory probability-distributions
asked Jan 19 at 16:26
LauraLaura
3018
$endgroup$
$begingroup$
If $Y=X^2$, none of the PDFs $f_{X,Y}$, $f_{Ymid X=x}$ and $f_{Xmid Y=y}$ 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
1
$begingroup$
@Did Exactly! But I will need the $f(x,y)$ distribution, right? How can I find her?
$endgroup$
– Laura
Jan 19 at 17:58
1
$begingroup$
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
1
$begingroup$
@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
2 days ago
|
show 2 more comments
2
2
2
1
$begingroup$
$X$ 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: $f_{Y} = 1/2sqrt{y}$. 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?
probability probability-theory probability-distributions
asked Jan 19 at 16:26
LauraLaura
3018
$endgroup$
$X$ 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: $f_{Y} = 1/2sqrt{y}$. 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?
probability probability-theory probability-distributions
probability probability-theory probability-distributions
asked Jan 19 at 16:26
LauraLaura
3018
asked Jan 19 at 16:26
LauraLaura
3018
asked Jan 19 at 16:26
LauraLaura
3018
asked Jan 19 at 16:26
LauraLaura
3018
asked Jan 19 at 16:26
LauraLaura
3018
3018
$begingroup$
If $Y=X^2$, none of the PDFs $f_{X,Y}$, $f_{Ymid X=x}$ and $f_{Xmid Y=y}$ 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
1
$begingroup$
@Did Exactly! But I will need the $f(x,y)$ distribution, right? How can I find her?
$endgroup$
– Laura
Jan 19 at 17:58
1
$begingroup$
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
1
$begingroup$
@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
2 days ago
|
show 2 more comments
$begingroup$
If $Y=X^2$, none of the PDFs $f_{X,Y}$, $f_{Ymid X=x}$ and $f_{Xmid Y=y}$ 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
1
$begingroup$
@Did Exactly! But I will need the $f(x,y)$ distribution, right? How can I find her?
$endgroup$
– Laura
Jan 19 at 17:58
1
$begingroup$
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
1
$begingroup$
@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
2 days ago
$begingroup$
If $Y=X^2$, none of the PDFs $f_{X,Y}$, $f_{Ymid X=x}$ and $f_{Xmid Y=y}$ 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
$begingroup$
If $Y=X^2$, none of the PDFs $f_{X,Y}$, $f_{Ymid X=x}$ and $f_{Xmid Y=y}$ 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
1
1
$begingroup$
@Did Exactly! But I will need the $f(x,y)$ distribution, right? How can I find her?
$endgroup$
– Laura
Jan 19 at 17:58
$begingroup$
@Did Exactly! But I will need the $f(x,y)$ distribution, right? How can I find her?
$endgroup$
– Laura
Jan 19 at 17:58
1
1
$begingroup$
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
$begingroup$
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
$begingroup$
Laura you may want to start with Dirac Delta.
$endgroup$
– Lee David Chung Lin
Jan 20 at 10:42
1
1
$begingroup$
@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
2 days ago
$begingroup$
@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
2 days ago
|
show 2 more comments
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$begingroup$
If $Y=X^2$, none of the PDFs $f_{X,Y}$, $f_{Ymid X=x}$ and $f_{Xmid Y=y}$ 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
1
$begingroup$
@Did Exactly! But I will need the $f(x,y)$ distribution, right? How can I find her?
$endgroup$
– Laura
Jan 19 at 17:58
1
$begingroup$
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
1
$begingroup$
@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
2 days ago