Sum / difference of two Gumbel
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I would like your help to derive the distribution of the sum/difference of two random variables which are distributed as Gumbel.
Consider $X$ having Gumbel distribution with "long tail on the right" as defined here, location $gamma$ (Euler constant), and scale $beta_x>0$.
Consider $Y$ having Gumbel distribution with "long tail on the right" as defined here, location $gamma$ (Euler constant), and scale $beta_y>0$.
(1) What is the distribution of $X-Y$?
According to Wikipedia, if $beta_x=beta_y=beta$, then $X-Y$ is Logistic with location $gamma-gamma=0$ and scale $beta$.
Can we say something when $beta_xneq beta_y$?
(2) What is the distribution of $X+Y$? My thoughts:
(2.1) $X+Y=X-(-Y)$
(2.2) $-Y$ is Gumbel with location $gamma$ and scale $-beta_y$
(2.3) Hence ?
probability probability-theory probability-distributions random-variables
asked Dec 14 '18 at 14:49
STFSTF
431422
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add a comment |
$begingroup$
I would like your help to derive the distribution of the sum/difference of two random variables which are distributed as Gumbel.
Consider $X$ having Gumbel distribution with "long tail on the right" as defined here, location $gamma$ (Euler constant), and scale $beta_x>0$.
Consider $Y$ having Gumbel distribution with "long tail on the right" as defined here, location $gamma$ (Euler constant), and scale $beta_y>0$.
(1) What is the distribution of $X-Y$?
According to Wikipedia, if $beta_x=beta_y=beta$, then $X-Y$ is Logistic with location $gamma-gamma=0$ and scale $beta$.
Can we say something when $beta_xneq beta_y$?
(2) What is the distribution of $X+Y$? My thoughts:
(2.1) $X+Y=X-(-Y)$
(2.2) $-Y$ is Gumbel with location $gamma$ and scale $-beta_y$
(2.3) Hence ?
probability probability-theory probability-distributions random-variables
asked Dec 14 '18 at 14:49
STFSTF
431422
$endgroup$
$begingroup$
Sorry; I misread your question and thought the $a_i$ were locationsand $gamma$ a scale. Having reconsidered, my technique won't address the problem you asked about, so I've deleted my answer.
$endgroup$
– J.G.
Dec 14 '18 at 17:19
$begingroup$
Thanks, I've reformulated my question, I hope the notation is less confusing now.
$endgroup$
– STF
Dec 14 '18 at 17:27
add a comment |
$begingroup$
I would like your help to derive the distribution of the sum/difference of two random variables which are distributed as Gumbel.
Consider $X$ having Gumbel distribution with "long tail on the right" as defined here, location $gamma$ (Euler constant), and scale $beta_x>0$.
Consider $Y$ having Gumbel distribution with "long tail on the right" as defined here, location $gamma$ (Euler constant), and scale $beta_y>0$.
(1) What is the distribution of $X-Y$?
According to Wikipedia, if $beta_x=beta_y=beta$, then $X-Y$ is Logistic with location $gamma-gamma=0$ and scale $beta$.
Can we say something when $beta_xneq beta_y$?
(2) What is the distribution of $X+Y$? My thoughts:
(2.1) $X+Y=X-(-Y)$
(2.2) $-Y$ is Gumbel with location $gamma$ and scale $-beta_y$
(2.3) Hence ?
probability probability-theory probability-distributions random-variables
asked Dec 14 '18 at 14:49
STFSTF
431422
$endgroup$
I would like your help to derive the distribution of the sum/difference of two random variables which are distributed as Gumbel.
Consider $X$ having Gumbel distribution with "long tail on the right" as defined here, location $gamma$ (Euler constant), and scale $beta_x>0$.
Consider $Y$ having Gumbel distribution with "long tail on the right" as defined here, location $gamma$ (Euler constant), and scale $beta_y>0$.
(1) What is the distribution of $X-Y$?
According to Wikipedia, if $beta_x=beta_y=beta$, then $X-Y$ is Logistic with location $gamma-gamma=0$ and scale $beta$.
Can we say something when $beta_xneq beta_y$?
(2) What is the distribution of $X+Y$? My thoughts:
(2.1) $X+Y=X-(-Y)$
(2.2) $-Y$ is Gumbel with location $gamma$ and scale $-beta_y$
(2.3) Hence ?
probability probability-theory probability-distributions random-variables
probability probability-theory probability-distributions random-variables
asked Dec 14 '18 at 14:49
STFSTF
431422
asked Dec 14 '18 at 14:49
STFSTF
431422
edited Dec 14 '18 at 17:26
STF
asked Dec 14 '18 at 14:49
STFSTF
431422
asked Dec 14 '18 at 14:49
STFSTF
431422
asked Dec 14 '18 at 14:49
STFSTF
431422
431422
$begingroup$
Sorry; I misread your question and thought the $a_i$ were locationsand $gamma$ a scale. Having reconsidered, my technique won't address the problem you asked about, so I've deleted my answer.
$endgroup$
– J.G.
Dec 14 '18 at 17:19
$begingroup$
Thanks, I've reformulated my question, I hope the notation is less confusing now.
$endgroup$
– STF
Dec 14 '18 at 17:27
add a comment |
$begingroup$
Sorry; I misread your question and thought the $a_i$ were locationsand $gamma$ a scale. Having reconsidered, my technique won't address the problem you asked about, so I've deleted my answer.
$endgroup$
– J.G.
Dec 14 '18 at 17:19
$begingroup$
Thanks, I've reformulated my question, I hope the notation is less confusing now.
$endgroup$
– STF
Dec 14 '18 at 17:27
$begingroup$
Sorry; I misread your question and thought the $a_i$ were locationsand $gamma$ a scale. Having reconsidered, my technique won't address the problem you asked about, so I've deleted my answer.
$endgroup$
– J.G.
Dec 14 '18 at 17:19
$begingroup$
Sorry; I misread your question and thought the $a_i$ were locationsand $gamma$ a scale. Having reconsidered, my technique won't address the problem you asked about, so I've deleted my answer.
$endgroup$
– J.G.
Dec 14 '18 at 17:19
$begingroup$
Thanks, I've reformulated my question, I hope the notation is less confusing now.
$endgroup$
– STF
Dec 14 '18 at 17:27
$begingroup$
Thanks, I've reformulated my question, I hope the notation is less confusing now.
$endgroup$
– STF
Dec 14 '18 at 17:27
add a comment |
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$begingroup$
Sorry; I misread your question and thought the $a_i$ were locationsand $gamma$ a scale. Having reconsidered, my technique won't address the problem you asked about, so I've deleted my answer.
$endgroup$
– J.G.
Dec 14 '18 at 17:19
$begingroup$
Thanks, I've reformulated my question, I hope the notation is less confusing now.
$endgroup$
– STF
Dec 14 '18 at 17:27