For which distribution same pdf is generated for given random variable
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For which of the distribution same pdf is generated for random variable X and 1/X.
Is it F(2,2)
statistics
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For which of the distribution same pdf is generated for random variable X and 1/X.
Is it F(2,2)
statistics
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5
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It is unclear from what you are actually asking.
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– 1123581321
May 27 at 8:10
1
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@1123581321 Suppose $X$ had density function $f(x)$ and $Y=frac1X$ with density function $g(y)$. The question wants to consider cases where $g(y) = f(y)$
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– Henry
May 27 at 9:03
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$begingroup$
For which of the distribution same pdf is generated for random variable X and 1/X.
Is it F(2,2)
statistics
$endgroup$
For which of the distribution same pdf is generated for random variable X and 1/X.
Is it F(2,2)
statistics
statistics
edited May 27 at 8:13
ten do
asked May 27 at 8:08
ten doten do
415 bronze badges
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5
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It is unclear from what you are actually asking.
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– 1123581321
May 27 at 8:10
1
$begingroup$
@1123581321 Suppose $X$ had density function $f(x)$ and $Y=frac1X$ with density function $g(y)$. The question wants to consider cases where $g(y) = f(y)$
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– Henry
May 27 at 9:03
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5
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It is unclear from what you are actually asking.
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– 1123581321
May 27 at 8:10
1
$begingroup$
@1123581321 Suppose $X$ had density function $f(x)$ and $Y=frac1X$ with density function $g(y)$. The question wants to consider cases where $g(y) = f(y)$
$endgroup$
– Henry
May 27 at 9:03
5
5
$begingroup$
It is unclear from what you are actually asking.
$endgroup$
– 1123581321
May 27 at 8:10
$begingroup$
It is unclear from what you are actually asking.
$endgroup$
– 1123581321
May 27 at 8:10
1
1
$begingroup$
@1123581321 Suppose $X$ had density function $f(x)$ and $Y=frac1X$ with density function $g(y)$. The question wants to consider cases where $g(y) = f(y)$
$endgroup$
– Henry
May 27 at 9:03
$begingroup$
@1123581321 Suppose $X$ had density function $f(x)$ and $Y=frac1X$ with density function $g(y)$. The question wants to consider cases where $g(y) = f(y)$
$endgroup$
– Henry
May 27 at 9:03
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2 Answers
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Let $U,V$ be iid random variables with $P(U=0)=0$.
Then if $X$ is defined as $frac{U}{V}$ it will have the same distribution as $frac1{X}=frac{V}{U}$.
If moreover $X$ has a PDF then $frac1{X}$ will also have the same PDF.
Special case: $U$ has chi-squared distribution. Then $X$ has $F$-distribution.
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It would be interesting if all solutions could be decomposed into a ratio like this
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– Henry
May 27 at 9:23
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@Henry Indeed. Uptil now I did not manage to prove that.
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– drhab
May 27 at 9:30
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@Rahul : Ah. Slow about an obvious. <sigh> Thanks.
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– Eric Towers
May 27 at 17:33
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You are looking for a solution to $f(x) = dfrac{f(1/x)}{x^2}$
So take any non-negative function $g(x)$ on $[-1,1]$ where $k= intlimits_{-1}^{1} g(x), dx$ is positive and finite
then a solution will be
- $f(x) = dfrac{g(x)}{2k} text { when } -1 le x le 1$
- $f(x) = dfrac{g(1/x)}{2kx^2} text { when } x lt -1 text{ or } x gt 1$
and I think all solutions will essentially be of this form
One solution is $f(x)=dfrac{1}{(1+x)^2}$ for $x gt 0$ and this is indeed an $F(2,2)$ distribution
but there are many others, including the Cauchy density $f(x)=dfrac{1}{pi(1+x^2)}$ on $x in mathbb R$
Another simple illustration, with $g(x)=1$ and so $k=2$, has $f(x)=frac14$ when $x in [-1,1]$ and $f(x)=frac1{4x^2}$ otherwise.
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2 Answers
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2 Answers
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$begingroup$
Let $U,V$ be iid random variables with $P(U=0)=0$.
Then if $X$ is defined as $frac{U}{V}$ it will have the same distribution as $frac1{X}=frac{V}{U}$.
If moreover $X$ has a PDF then $frac1{X}$ will also have the same PDF.
Special case: $U$ has chi-squared distribution. Then $X$ has $F$-distribution.
$endgroup$
$begingroup$
It would be interesting if all solutions could be decomposed into a ratio like this
$endgroup$
– Henry
May 27 at 9:23
$begingroup$
@Henry Indeed. Uptil now I did not manage to prove that.
$endgroup$
– drhab
May 27 at 9:30
$begingroup$
@Rahul : Ah. Slow about an obvious. <sigh> Thanks.
$endgroup$
– Eric Towers
May 27 at 17:33
add a comment
|
$begingroup$
Let $U,V$ be iid random variables with $P(U=0)=0$.
Then if $X$ is defined as $frac{U}{V}$ it will have the same distribution as $frac1{X}=frac{V}{U}$.
If moreover $X$ has a PDF then $frac1{X}$ will also have the same PDF.
Special case: $U$ has chi-squared distribution. Then $X$ has $F$-distribution.
$endgroup$
$begingroup$
It would be interesting if all solutions could be decomposed into a ratio like this
$endgroup$
– Henry
May 27 at 9:23
$begingroup$
@Henry Indeed. Uptil now I did not manage to prove that.
$endgroup$
– drhab
May 27 at 9:30
$begingroup$
@Rahul : Ah. Slow about an obvious. <sigh> Thanks.
$endgroup$
– Eric Towers
May 27 at 17:33
add a comment
|
$begingroup$
Let $U,V$ be iid random variables with $P(U=0)=0$.
Then if $X$ is defined as $frac{U}{V}$ it will have the same distribution as $frac1{X}=frac{V}{U}$.
If moreover $X$ has a PDF then $frac1{X}$ will also have the same PDF.
Special case: $U$ has chi-squared distribution. Then $X$ has $F$-distribution.
$endgroup$
Let $U,V$ be iid random variables with $P(U=0)=0$.
Then if $X$ is defined as $frac{U}{V}$ it will have the same distribution as $frac1{X}=frac{V}{U}$.
If moreover $X$ has a PDF then $frac1{X}$ will also have the same PDF.
Special case: $U$ has chi-squared distribution. Then $X$ has $F$-distribution.
answered May 27 at 9:00
drhabdrhab
114k5 gold badges51 silver badges143 bronze badges
114k5 gold badges51 silver badges143 bronze badges
$begingroup$
It would be interesting if all solutions could be decomposed into a ratio like this
$endgroup$
– Henry
May 27 at 9:23
$begingroup$
@Henry Indeed. Uptil now I did not manage to prove that.
$endgroup$
– drhab
May 27 at 9:30
$begingroup$
@Rahul : Ah. Slow about an obvious. <sigh> Thanks.
$endgroup$
– Eric Towers
May 27 at 17:33
add a comment
|
$begingroup$
It would be interesting if all solutions could be decomposed into a ratio like this
$endgroup$
– Henry
May 27 at 9:23
$begingroup$
@Henry Indeed. Uptil now I did not manage to prove that.
$endgroup$
– drhab
May 27 at 9:30
$begingroup$
@Rahul : Ah. Slow about an obvious. <sigh> Thanks.
$endgroup$
– Eric Towers
May 27 at 17:33
$begingroup$
It would be interesting if all solutions could be decomposed into a ratio like this
$endgroup$
– Henry
May 27 at 9:23
$begingroup$
It would be interesting if all solutions could be decomposed into a ratio like this
$endgroup$
– Henry
May 27 at 9:23
$begingroup$
@Henry Indeed. Uptil now I did not manage to prove that.
$endgroup$
– drhab
May 27 at 9:30
$begingroup$
@Henry Indeed. Uptil now I did not manage to prove that.
$endgroup$
– drhab
May 27 at 9:30
$begingroup$
@Rahul : Ah. Slow about an obvious. <sigh> Thanks.
$endgroup$
– Eric Towers
May 27 at 17:33
$begingroup$
@Rahul : Ah. Slow about an obvious. <sigh> Thanks.
$endgroup$
– Eric Towers
May 27 at 17:33
add a comment
|
$begingroup$
You are looking for a solution to $f(x) = dfrac{f(1/x)}{x^2}$
So take any non-negative function $g(x)$ on $[-1,1]$ where $k= intlimits_{-1}^{1} g(x), dx$ is positive and finite
then a solution will be
- $f(x) = dfrac{g(x)}{2k} text { when } -1 le x le 1$
- $f(x) = dfrac{g(1/x)}{2kx^2} text { when } x lt -1 text{ or } x gt 1$
and I think all solutions will essentially be of this form
One solution is $f(x)=dfrac{1}{(1+x)^2}$ for $x gt 0$ and this is indeed an $F(2,2)$ distribution
but there are many others, including the Cauchy density $f(x)=dfrac{1}{pi(1+x^2)}$ on $x in mathbb R$
Another simple illustration, with $g(x)=1$ and so $k=2$, has $f(x)=frac14$ when $x in [-1,1]$ and $f(x)=frac1{4x^2}$ otherwise.
$endgroup$
add a comment
|
$begingroup$
You are looking for a solution to $f(x) = dfrac{f(1/x)}{x^2}$
So take any non-negative function $g(x)$ on $[-1,1]$ where $k= intlimits_{-1}^{1} g(x), dx$ is positive and finite
then a solution will be
- $f(x) = dfrac{g(x)}{2k} text { when } -1 le x le 1$
- $f(x) = dfrac{g(1/x)}{2kx^2} text { when } x lt -1 text{ or } x gt 1$
and I think all solutions will essentially be of this form
One solution is $f(x)=dfrac{1}{(1+x)^2}$ for $x gt 0$ and this is indeed an $F(2,2)$ distribution
but there are many others, including the Cauchy density $f(x)=dfrac{1}{pi(1+x^2)}$ on $x in mathbb R$
Another simple illustration, with $g(x)=1$ and so $k=2$, has $f(x)=frac14$ when $x in [-1,1]$ and $f(x)=frac1{4x^2}$ otherwise.
$endgroup$
add a comment
|
$begingroup$
You are looking for a solution to $f(x) = dfrac{f(1/x)}{x^2}$
So take any non-negative function $g(x)$ on $[-1,1]$ where $k= intlimits_{-1}^{1} g(x), dx$ is positive and finite
then a solution will be
- $f(x) = dfrac{g(x)}{2k} text { when } -1 le x le 1$
- $f(x) = dfrac{g(1/x)}{2kx^2} text { when } x lt -1 text{ or } x gt 1$
and I think all solutions will essentially be of this form
One solution is $f(x)=dfrac{1}{(1+x)^2}$ for $x gt 0$ and this is indeed an $F(2,2)$ distribution
but there are many others, including the Cauchy density $f(x)=dfrac{1}{pi(1+x^2)}$ on $x in mathbb R$
Another simple illustration, with $g(x)=1$ and so $k=2$, has $f(x)=frac14$ when $x in [-1,1]$ and $f(x)=frac1{4x^2}$ otherwise.
$endgroup$
You are looking for a solution to $f(x) = dfrac{f(1/x)}{x^2}$
So take any non-negative function $g(x)$ on $[-1,1]$ where $k= intlimits_{-1}^{1} g(x), dx$ is positive and finite
then a solution will be
- $f(x) = dfrac{g(x)}{2k} text { when } -1 le x le 1$
- $f(x) = dfrac{g(1/x)}{2kx^2} text { when } x lt -1 text{ or } x gt 1$
and I think all solutions will essentially be of this form
One solution is $f(x)=dfrac{1}{(1+x)^2}$ for $x gt 0$ and this is indeed an $F(2,2)$ distribution
but there are many others, including the Cauchy density $f(x)=dfrac{1}{pi(1+x^2)}$ on $x in mathbb R$
Another simple illustration, with $g(x)=1$ and so $k=2$, has $f(x)=frac14$ when $x in [-1,1]$ and $f(x)=frac1{4x^2}$ otherwise.
edited May 27 at 18:05
answered May 27 at 8:59
HenryHenry
107k4 gold badges86 silver badges177 bronze badges
107k4 gold badges86 silver badges177 bronze badges
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$begingroup$
It is unclear from what you are actually asking.
$endgroup$
– 1123581321
May 27 at 8:10
1
$begingroup$
@1123581321 Suppose $X$ had density function $f(x)$ and $Y=frac1X$ with density function $g(y)$. The question wants to consider cases where $g(y) = f(y)$
$endgroup$
– Henry
May 27 at 9:03