What is the distribution of $aX$ when $X$ is Gumbel? What is the distribution of $-X$ when $X$ is Gumbel?
$begingroup$
If $X$ is a random variable distributed as a Gumbel with location $gamma$ and scale $1$ (where $gamma$ is the Euler constant)
[Hence the mean is
$$
gamma-1timesgamma=0
$$
and the variance is
$$
pi^2/6times 1^2=pi^2/6
$$]
(1) What is the distribution of $aX$ where $ain mathbb{R}$? $a$ should be $>0$?
(2) What is the distribution of $-X$?
I don't know if it is relevant, but here I'm using the definition of Gumbel as in Mathworld and Matlab (sign flipped with respect to the definition of Gumbel in Wikipedia)
probability probability-theory probability-distributions random-variables
asked Dec 14 '18 at 11:59
STFSTF
111420
$endgroup$
add a comment |
$begingroup$
If $X$ is a random variable distributed as a Gumbel with location $gamma$ and scale $1$ (where $gamma$ is the Euler constant)
[Hence the mean is
$$
gamma-1timesgamma=0
$$
and the variance is
$$
pi^2/6times 1^2=pi^2/6
$$]
(1) What is the distribution of $aX$ where $ain mathbb{R}$? $a$ should be $>0$?
(2) What is the distribution of $-X$?
I don't know if it is relevant, but here I'm using the definition of Gumbel as in Mathworld and Matlab (sign flipped with respect to the definition of Gumbel in Wikipedia)
probability probability-theory probability-distributions random-variables
asked Dec 14 '18 at 11:59
STFSTF
111420
$endgroup$
$begingroup$
Please show us what you have tried so far.
$endgroup$
– Stockfish
Dec 14 '18 at 12:14
$begingroup$
(1) I think that $aX$ is Gumbel with location $gamma$ and scale that should be adjusted. Maybe the new scale is simply $a$? (2) The Gumbel is not symmetric around zero. So I'm tempted to say that $-X$ is not Gumbel But I don't know what else can be said.
$endgroup$
– STF
Dec 14 '18 at 12:17
add a comment |
$begingroup$
If $X$ is a random variable distributed as a Gumbel with location $gamma$ and scale $1$ (where $gamma$ is the Euler constant)
[Hence the mean is
$$
gamma-1timesgamma=0
$$
and the variance is
$$
pi^2/6times 1^2=pi^2/6
$$]
(1) What is the distribution of $aX$ where $ain mathbb{R}$? $a$ should be $>0$?
(2) What is the distribution of $-X$?
I don't know if it is relevant, but here I'm using the definition of Gumbel as in Mathworld and Matlab (sign flipped with respect to the definition of Gumbel in Wikipedia)
probability probability-theory probability-distributions random-variables
asked Dec 14 '18 at 11:59
STFSTF
111420
$endgroup$
If $X$ is a random variable distributed as a Gumbel with location $gamma$ and scale $1$ (where $gamma$ is the Euler constant)
[Hence the mean is
$$
gamma-1timesgamma=0
$$
and the variance is
$$
pi^2/6times 1^2=pi^2/6
$$]
(1) What is the distribution of $aX$ where $ain mathbb{R}$? $a$ should be $>0$?
(2) What is the distribution of $-X$?
I don't know if it is relevant, but here I'm using the definition of Gumbel as in Mathworld and Matlab (sign flipped with respect to the definition of Gumbel in Wikipedia)
probability probability-theory probability-distributions random-variables
probability probability-theory probability-distributions random-variables
asked Dec 14 '18 at 11:59
STFSTF
111420
asked Dec 14 '18 at 11:59
STFSTF
111420
edited Dec 14 '18 at 12:28
STF
asked Dec 14 '18 at 11:59
STFSTF
111420
asked Dec 14 '18 at 11:59
STFSTF
111420
asked Dec 14 '18 at 11:59
STFSTF
111420
111420
$begingroup$
Please show us what you have tried so far.
$endgroup$
– Stockfish
Dec 14 '18 at 12:14
$begingroup$
(1) I think that $aX$ is Gumbel with location $gamma$ and scale that should be adjusted. Maybe the new scale is simply $a$? (2) The Gumbel is not symmetric around zero. So I'm tempted to say that $-X$ is not Gumbel But I don't know what else can be said.
$endgroup$
– STF
Dec 14 '18 at 12:17
add a comment |
$begingroup$
Please show us what you have tried so far.
$endgroup$
– Stockfish
Dec 14 '18 at 12:14
$begingroup$
(1) I think that $aX$ is Gumbel with location $gamma$ and scale that should be adjusted. Maybe the new scale is simply $a$? (2) The Gumbel is not symmetric around zero. So I'm tempted to say that $-X$ is not Gumbel But I don't know what else can be said.
$endgroup$
– STF
Dec 14 '18 at 12:17
$begingroup$
Please show us what you have tried so far.
$endgroup$
– Stockfish
Dec 14 '18 at 12:14
$begingroup$
Please show us what you have tried so far.
$endgroup$
– Stockfish
Dec 14 '18 at 12:14
$begingroup$
(1) I think that $aX$ is Gumbel with location $gamma$ and scale that should be adjusted. Maybe the new scale is simply $a$? (2) The Gumbel is not symmetric around zero. So I'm tempted to say that $-X$ is not Gumbel But I don't know what else can be said.
$endgroup$
– STF
Dec 14 '18 at 12:17
$begingroup$
(1) I think that $aX$ is Gumbel with location $gamma$ and scale that should be adjusted. Maybe the new scale is simply $a$? (2) The Gumbel is not symmetric around zero. So I'm tempted to say that $-X$ is not Gumbel But I don't know what else can be said.
$endgroup$
– STF
Dec 14 '18 at 12:17
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Still a Gumbel distribution.
As with Normal distributions, the family of Gumbel distributions includes all pdfs of the form $|beta|^{-1}fleft(frac{x-mu}{beta}right)$, where $f$ is in the family. So $Xmapsto aX$ just scales $beta$ to $abeta$, including in the case $a=-1$.
edited Dec 16 '18 at 15:34
Davide Giraudo
126k16150261
answered Dec 14 '18 at 12:16
J.G.J.G.
26.1k22539
$endgroup$
$begingroup$
Thanks. But for $a=-1$ then the new scale would be $-beta$ which is $<0$ (which is not possible, right?). Shouldn't we impose $a>0$?
$endgroup$
– STF
Dec 14 '18 at 12:19
$begingroup$
What about point (2) of my question?
$endgroup$
– STF
Dec 14 '18 at 12:19
$begingroup$
@STF (2) is, as explained, a special case of (1). You shouldn't worry about $beta<0$ as long as you see the sign of $beta$ as saying whether it's a left- or right-handed Gumbel distribution. This is an area where the situation is very different from the Normal distribution, because the pdf isn't even.
$endgroup$
– J.G.
Dec 14 '18 at 12:55
$begingroup$
could you given a look to this other question of mine if you have some time? Thanks math.stackexchange.com/questions/3039451/…
$endgroup$
– STF
Dec 14 '18 at 16:26
add a comment |
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1 Answer
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1 Answer
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oldest
votes
$begingroup$
Still a Gumbel distribution.
As with Normal distributions, the family of Gumbel distributions includes all pdfs of the form $|beta|^{-1}fleft(frac{x-mu}{beta}right)$, where $f$ is in the family. So $Xmapsto aX$ just scales $beta$ to $abeta$, including in the case $a=-1$.
edited Dec 16 '18 at 15:34
Davide Giraudo
126k16150261
answered Dec 14 '18 at 12:16
J.G.J.G.
26.1k22539
$endgroup$
$begingroup$
Thanks. But for $a=-1$ then the new scale would be $-beta$ which is $<0$ (which is not possible, right?). Shouldn't we impose $a>0$?
$endgroup$
– STF
Dec 14 '18 at 12:19
$begingroup$
What about point (2) of my question?
$endgroup$
– STF
Dec 14 '18 at 12:19
$begingroup$
@STF (2) is, as explained, a special case of (1). You shouldn't worry about $beta<0$ as long as you see the sign of $beta$ as saying whether it's a left- or right-handed Gumbel distribution. This is an area where the situation is very different from the Normal distribution, because the pdf isn't even.
$endgroup$
– J.G.
Dec 14 '18 at 12:55
$begingroup$
could you given a look to this other question of mine if you have some time? Thanks math.stackexchange.com/questions/3039451/…
$endgroup$
– STF
Dec 14 '18 at 16:26
add a comment |
$begingroup$
Still a Gumbel distribution.
As with Normal distributions, the family of Gumbel distributions includes all pdfs of the form $|beta|^{-1}fleft(frac{x-mu}{beta}right)$, where $f$ is in the family. So $Xmapsto aX$ just scales $beta$ to $abeta$, including in the case $a=-1$.
edited Dec 16 '18 at 15:34
Davide Giraudo
126k16150261
answered Dec 14 '18 at 12:16
J.G.J.G.
26.1k22539
$endgroup$
$begingroup$
Thanks. But for $a=-1$ then the new scale would be $-beta$ which is $<0$ (which is not possible, right?). Shouldn't we impose $a>0$?
$endgroup$
– STF
Dec 14 '18 at 12:19
$begingroup$
What about point (2) of my question?
$endgroup$
– STF
Dec 14 '18 at 12:19
$begingroup$
@STF (2) is, as explained, a special case of (1). You shouldn't worry about $beta<0$ as long as you see the sign of $beta$ as saying whether it's a left- or right-handed Gumbel distribution. This is an area where the situation is very different from the Normal distribution, because the pdf isn't even.
$endgroup$
– J.G.
Dec 14 '18 at 12:55
$begingroup$
could you given a look to this other question of mine if you have some time? Thanks math.stackexchange.com/questions/3039451/…
$endgroup$
– STF
Dec 14 '18 at 16:26
add a comment |
$begingroup$
Still a Gumbel distribution.
As with Normal distributions, the family of Gumbel distributions includes all pdfs of the form $|beta|^{-1}fleft(frac{x-mu}{beta}right)$, where $f$ is in the family. So $Xmapsto aX$ just scales $beta$ to $abeta$, including in the case $a=-1$.
edited Dec 16 '18 at 15:34
Davide Giraudo
126k16150261
answered Dec 14 '18 at 12:16
J.G.J.G.
26.1k22539
$endgroup$
Still a Gumbel distribution.
As with Normal distributions, the family of Gumbel distributions includes all pdfs of the form $|beta|^{-1}fleft(frac{x-mu}{beta}right)$, where $f$ is in the family. So $Xmapsto aX$ just scales $beta$ to $abeta$, including in the case $a=-1$.
edited Dec 16 '18 at 15:34
Davide Giraudo
126k16150261
answered Dec 14 '18 at 12:16
J.G.J.G.
26.1k22539
edited Dec 16 '18 at 15:34
Davide Giraudo
126k16150261
edited Dec 16 '18 at 15:34
Davide Giraudo
126k16150261
edited Dec 16 '18 at 15:34
Davide Giraudo
126k16150261
126k16150261
answered Dec 14 '18 at 12:16
J.G.J.G.
26.1k22539
answered Dec 14 '18 at 12:16
J.G.J.G.
26.1k22539
answered Dec 14 '18 at 12:16
J.G.J.G.
26.1k22539
26.1k22539
$begingroup$
Thanks. But for $a=-1$ then the new scale would be $-beta$ which is $<0$ (which is not possible, right?). Shouldn't we impose $a>0$?
$endgroup$
– STF
Dec 14 '18 at 12:19
$begingroup$
What about point (2) of my question?
$endgroup$
– STF
Dec 14 '18 at 12:19
$begingroup$
@STF (2) is, as explained, a special case of (1). You shouldn't worry about $beta<0$ as long as you see the sign of $beta$ as saying whether it's a left- or right-handed Gumbel distribution. This is an area where the situation is very different from the Normal distribution, because the pdf isn't even.
$endgroup$
– J.G.
Dec 14 '18 at 12:55
$begingroup$
could you given a look to this other question of mine if you have some time? Thanks math.stackexchange.com/questions/3039451/…
$endgroup$
– STF
Dec 14 '18 at 16:26
add a comment |
$begingroup$
Thanks. But for $a=-1$ then the new scale would be $-beta$ which is $<0$ (which is not possible, right?). Shouldn't we impose $a>0$?
$endgroup$
– STF
Dec 14 '18 at 12:19
$begingroup$
What about point (2) of my question?
$endgroup$
– STF
Dec 14 '18 at 12:19
$begingroup$
@STF (2) is, as explained, a special case of (1). You shouldn't worry about $beta<0$ as long as you see the sign of $beta$ as saying whether it's a left- or right-handed Gumbel distribution. This is an area where the situation is very different from the Normal distribution, because the pdf isn't even.
$endgroup$
– J.G.
Dec 14 '18 at 12:55
$begingroup$
could you given a look to this other question of mine if you have some time? Thanks math.stackexchange.com/questions/3039451/…
$endgroup$
– STF
Dec 14 '18 at 16:26
$begingroup$
Thanks. But for $a=-1$ then the new scale would be $-beta$ which is $<0$ (which is not possible, right?). Shouldn't we impose $a>0$?
$endgroup$
– STF
Dec 14 '18 at 12:19
$begingroup$
Thanks. But for $a=-1$ then the new scale would be $-beta$ which is $<0$ (which is not possible, right?). Shouldn't we impose $a>0$?
$endgroup$
– STF
Dec 14 '18 at 12:19
$begingroup$
What about point (2) of my question?
$endgroup$
– STF
Dec 14 '18 at 12:19
$begingroup$
What about point (2) of my question?
$endgroup$
– STF
Dec 14 '18 at 12:19
$begingroup$
@STF (2) is, as explained, a special case of (1). You shouldn't worry about $beta<0$ as long as you see the sign of $beta$ as saying whether it's a left- or right-handed Gumbel distribution. This is an area where the situation is very different from the Normal distribution, because the pdf isn't even.
$endgroup$
– J.G.
Dec 14 '18 at 12:55
$begingroup$
@STF (2) is, as explained, a special case of (1). You shouldn't worry about $beta<0$ as long as you see the sign of $beta$ as saying whether it's a left- or right-handed Gumbel distribution. This is an area where the situation is very different from the Normal distribution, because the pdf isn't even.
$endgroup$
– J.G.
Dec 14 '18 at 12:55
$begingroup$
could you given a look to this other question of mine if you have some time? Thanks math.stackexchange.com/questions/3039451/…
$endgroup$
– STF
Dec 14 '18 at 16:26
$begingroup$
could you given a look to this other question of mine if you have some time? Thanks math.stackexchange.com/questions/3039451/…
$endgroup$
– STF
Dec 14 '18 at 16:26
add a comment |
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$begingroup$
Please show us what you have tried so far.
$endgroup$
– Stockfish
Dec 14 '18 at 12:14
$begingroup$
(1) I think that $aX$ is Gumbel with location $gamma$ and scale that should be adjusted. Maybe the new scale is simply $a$? (2) The Gumbel is not symmetric around zero. So I'm tempted to say that $-X$ is not Gumbel But I don't know what else can be said.
$endgroup$
– STF
Dec 14 '18 at 12:17