Probability distribution of independent random variables [closed]
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X,Y are independent random variables with distribution N(0,1).
Find probability distribution of random variable $frac{Y}{|X|}$.
probability
asked Jan 9 at 13:52
JohnJohn
42
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closed as off-topic by 5xum, StubbornAtom, verret, Math1000, Lord_Farin Jan 9 at 23:00
This question appears to be off-topic. The users who voted to close gave this specific reason:
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If this question can be reworded to fit the rules in the help center, please edit the question.
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$begingroup$
X,Y are independent random variables with distribution N(0,1).
Find probability distribution of random variable $frac{Y}{|X|}$.
probability
asked Jan 9 at 13:52
JohnJohn
42
$endgroup$
closed as off-topic by 5xum, StubbornAtom, verret, Math1000, Lord_Farin Jan 9 at 23:00
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – 5xum, StubbornAtom, verret, Math1000, Lord_Farin
If this question can be reworded to fit the rules in the help center, please edit the question.
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You may get better answers by following the advice in math.stackexchange.com/help/how-to-ask and by trying to ask in a way that is more like questions on this site that got good responses. One way is to show what you do and don't understand and what you tried. "Show" means actually write out your work. Note, the way to fix this is by editing the question, not by putting the information in comments.
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– David K
Jan 9 at 14:05
add a comment |
$begingroup$
X,Y are independent random variables with distribution N(0,1).
Find probability distribution of random variable $frac{Y}{|X|}$.
probability
asked Jan 9 at 13:52
JohnJohn
42
$endgroup$
X,Y are independent random variables with distribution N(0,1).
Find probability distribution of random variable $frac{Y}{|X|}$.
probability
probability
asked Jan 9 at 13:52
JohnJohn
42
asked Jan 9 at 13:52
JohnJohn
42
asked Jan 9 at 13:52
JohnJohn
42
asked Jan 9 at 13:52
JohnJohn
42
asked Jan 9 at 13:52
JohnJohn
42
42
closed as off-topic by 5xum, StubbornAtom, verret, Math1000, Lord_Farin Jan 9 at 23:00
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – 5xum, StubbornAtom, verret, Math1000, Lord_Farin
If this question can be reworded to fit the rules in the help center, please edit the question.
closed as off-topic by 5xum, StubbornAtom, verret, Math1000, Lord_Farin Jan 9 at 23:00
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – 5xum, StubbornAtom, verret, Math1000, Lord_Farin
If this question can be reworded to fit the rules in the help center, please edit the question.
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You may get better answers by following the advice in math.stackexchange.com/help/how-to-ask and by trying to ask in a way that is more like questions on this site that got good responses. One way is to show what you do and don't understand and what you tried. "Show" means actually write out your work. Note, the way to fix this is by editing the question, not by putting the information in comments.
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– David K
Jan 9 at 14:05
add a comment |
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You may get better answers by following the advice in math.stackexchange.com/help/how-to-ask and by trying to ask in a way that is more like questions on this site that got good responses. One way is to show what you do and don't understand and what you tried. "Show" means actually write out your work. Note, the way to fix this is by editing the question, not by putting the information in comments.
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– David K
Jan 9 at 14:05
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You may get better answers by following the advice in math.stackexchange.com/help/how-to-ask and by trying to ask in a way that is more like questions on this site that got good responses. One way is to show what you do and don't understand and what you tried. "Show" means actually write out your work. Note, the way to fix this is by editing the question, not by putting the information in comments.
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– David K
Jan 9 at 14:05
$begingroup$
You may get better answers by following the advice in math.stackexchange.com/help/how-to-ask and by trying to ask in a way that is more like questions on this site that got good responses. One way is to show what you do and don't understand and what you tried. "Show" means actually write out your work. Note, the way to fix this is by editing the question, not by putting the information in comments.
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– David K
Jan 9 at 14:05
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1 Answer
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As noted here, there's a formula for the pdf of a ratio of continuous random variables. In particular $Z:=A/B$ with $A,,B$ independent has pdf $int_{Bbb R}|b|f_A(bz)f_B(b) db$. For our present purposes this is $$int_0^inftyfrac{x}{pi}exp-frac{(z^2+1)x^2}{2}dx=frac{1}{pi(z^2+1)}.$$In other words, $Y/|X|$ is Cauchy-distributed.
answered Jan 9 at 14:10
J.G.J.G.
24.2k22539
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
As noted here, there's a formula for the pdf of a ratio of continuous random variables. In particular $Z:=A/B$ with $A,,B$ independent has pdf $int_{Bbb R}|b|f_A(bz)f_B(b) db$. For our present purposes this is $$int_0^inftyfrac{x}{pi}exp-frac{(z^2+1)x^2}{2}dx=frac{1}{pi(z^2+1)}.$$In other words, $Y/|X|$ is Cauchy-distributed.
answered Jan 9 at 14:10
J.G.J.G.
24.2k22539
$endgroup$
add a comment |
$begingroup$
As noted here, there's a formula for the pdf of a ratio of continuous random variables. In particular $Z:=A/B$ with $A,,B$ independent has pdf $int_{Bbb R}|b|f_A(bz)f_B(b) db$. For our present purposes this is $$int_0^inftyfrac{x}{pi}exp-frac{(z^2+1)x^2}{2}dx=frac{1}{pi(z^2+1)}.$$In other words, $Y/|X|$ is Cauchy-distributed.
answered Jan 9 at 14:10
J.G.J.G.
24.2k22539
$endgroup$
add a comment |
$begingroup$
As noted here, there's a formula for the pdf of a ratio of continuous random variables. In particular $Z:=A/B$ with $A,,B$ independent has pdf $int_{Bbb R}|b|f_A(bz)f_B(b) db$. For our present purposes this is $$int_0^inftyfrac{x}{pi}exp-frac{(z^2+1)x^2}{2}dx=frac{1}{pi(z^2+1)}.$$In other words, $Y/|X|$ is Cauchy-distributed.
answered Jan 9 at 14:10
J.G.J.G.
24.2k22539
$endgroup$
As noted here, there's a formula for the pdf of a ratio of continuous random variables. In particular $Z:=A/B$ with $A,,B$ independent has pdf $int_{Bbb R}|b|f_A(bz)f_B(b) db$. For our present purposes this is $$int_0^inftyfrac{x}{pi}exp-frac{(z^2+1)x^2}{2}dx=frac{1}{pi(z^2+1)}.$$In other words, $Y/|X|$ is Cauchy-distributed.
answered Jan 9 at 14:10
J.G.J.G.
24.2k22539
answered Jan 9 at 14:10
J.G.J.G.
24.2k22539
answered Jan 9 at 14:10
J.G.J.G.
24.2k22539
answered Jan 9 at 14:10
J.G.J.G.
24.2k22539
24.2k22539
add a comment |
add a comment |
$begingroup$
You may get better answers by following the advice in math.stackexchange.com/help/how-to-ask and by trying to ask in a way that is more like questions on this site that got good responses. One way is to show what you do and don't understand and what you tried. "Show" means actually write out your work. Note, the way to fix this is by editing the question, not by putting the information in comments.
$endgroup$
– David K
Jan 9 at 14:05