Show median zero and symmetry of differences of Gumbel [closed]

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Consider a random vector $epsilonequiv (epsilon_1, epsilon_2, epsilon_0)$. Suppose $epsilon_1, epsilon_2, epsilon_0$ are i.i.d., distributed as Gumbel with location $0$ and scale $1$.

Take $Vequiv (V_1, V_2, V_3)$ with
$$
V_1equiv epsilon_1-epsilon_0
$$
$$
V_2equiv epsilon_2-epsilon_0
$$
$$
V_3equiv V_1-V_2
$$

Could you help me to show - if true - that $forall j in 1,2,3$

1) The distribution of $V_j$ has median $0$

2) The distribution of $V_j$ is symmetric around zero

probability probability-theory probability-distributions random-variables extreme-value-analysis

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asked Jul 31 at 15:13

TEX

1919

closed as off-topic by heropup, Mostafa Ayaz, amWhy, Isaac Browne, Leucippus Aug 1 at 2:16

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 improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – heropup, Mostafa Ayaz, amWhy, Isaac Browne, Leucippus

If this question can be reworded to fit the rules in the help center, please edit the question.

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up vote
-1
down vote

favorite

Consider a random vector $epsilonequiv (epsilon_1, epsilon_2, epsilon_0)$. Suppose $epsilon_1, epsilon_2, epsilon_0$ are i.i.d., distributed as Gumbel with location $0$ and scale $1$.

Take $Vequiv (V_1, V_2, V_3)$ with
$$
V_1equiv epsilon_1-epsilon_0
$$
$$
V_2equiv epsilon_2-epsilon_0
$$
$$
V_3equiv V_1-V_2
$$

Could you help me to show - if true - that $forall j in 1,2,3$

1) The distribution of $V_j$ has median $0$

2) The distribution of $V_j$ is symmetric around zero

probability probability-theory probability-distributions random-variables extreme-value-analysis

share|cite|improve this question

asked Jul 31 at 15:13

TEX

1919

closed as off-topic by heropup, Mostafa Ayaz, amWhy, Isaac Browne, Leucippus Aug 1 at 2:16

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 improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – heropup, Mostafa Ayaz, amWhy, Isaac Browne, Leucippus

If this question can be reworded to fit the rules in the help center, please edit the question.

add a comment | 

up vote
-1
down vote

favorite

up vote
-1
down vote

favorite

Consider a random vector $epsilonequiv (epsilon_1, epsilon_2, epsilon_0)$. Suppose $epsilon_1, epsilon_2, epsilon_0$ are i.i.d., distributed as Gumbel with location $0$ and scale $1$.

Take $Vequiv (V_1, V_2, V_3)$ with
$$
V_1equiv epsilon_1-epsilon_0
$$
$$
V_2equiv epsilon_2-epsilon_0
$$
$$
V_3equiv V_1-V_2
$$

Could you help me to show - if true - that $forall j in 1,2,3$

1) The distribution of $V_j$ has median $0$

2) The distribution of $V_j$ is symmetric around zero

probability probability-theory probability-distributions random-variables extreme-value-analysis

share|cite|improve this question

asked Jul 31 at 15:13

TEX

1919

Consider a random vector $epsilonequiv (epsilon_1, epsilon_2, epsilon_0)$. Suppose $epsilon_1, epsilon_2, epsilon_0$ are i.i.d., distributed as Gumbel with location $0$ and scale $1$.

Take $Vequiv (V_1, V_2, V_3)$ with
$$
V_1equiv epsilon_1-epsilon_0
$$
$$
V_2equiv epsilon_2-epsilon_0
$$
$$
V_3equiv V_1-V_2
$$

Could you help me to show - if true - that $forall j in 1,2,3$

1) The distribution of $V_j$ has median $0$

2) The distribution of $V_j$ is symmetric around zero

probability probability-theory probability-distributions random-variables extreme-value-analysis

share|cite|improve this question

asked Jul 31 at 15:13

TEX

1919

share|cite|improve this question

share|cite|improve this question

asked Jul 31 at 15:13

TEX

1919

asked Jul 31 at 15:13

TEX

1919

asked Jul 31 at 15:13

TEX

1919

1919

closed as off-topic by heropup, Mostafa Ayaz, amWhy, Isaac Browne, Leucippus Aug 1 at 2:16

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 improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – heropup, Mostafa Ayaz, amWhy, Isaac Browne, Leucippus

If this question can be reworded to fit the rules in the help center, please edit the question.

closed as off-topic by heropup, Mostafa Ayaz, amWhy, Isaac Browne, Leucippus Aug 1 at 2:16

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 improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – heropup, Mostafa Ayaz, amWhy, Isaac Browne, Leucippus

If this question can be reworded to fit the rules in the help center, please edit the question.

add a comment | 

add a comment | 

2 Answers
2

active

oldest

votes

up vote
3
down vote



accepted

The Gumbel distribution is not important here. In general, whenever $X$ and $Y$ have the same distribution and are independent, then $X-Y$ will be symmetric about zero. This is because $X-Y$ has the same joint distribution as $Y-X$, so for all $cin mathbb R$,
$$
P(X-Yle c)=P(Y-Xle -c)stackrelX-Ystackreld=Y-X=P(X-Yle -c)
$$
which says $X-Y$ is symmetric about $0$. This immediately implies its median of $X-Y$ is $0$, since $P(X-Yle 0)=P(X-Yge 0$) and these probabilities sum to $1+P(X-Y=0)ge 1$, so they must both be at least $frac12$.

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answered Jul 31 at 17:38

Mike Earnest

14.7k11644

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up vote
2
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Use this property: if $X$ and $Y$ are distributed as Gumbel distribution $textGum(alpha, beta)$ then $X - Y sim textLog(0, beta)$ (see Logistic distribution) and since for logistic distribution has median equal to location parameter $mu = 0$ then the $textmed V_j = 0$. Logistic distribution is symmetric as well, so the second part of your question is also true.

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answered Jul 31 at 15:24

pointguard0

689517

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2 Answers
2

active

oldest

votes

2 Answers
2

active

oldest

votes

active

oldest

votes

active

oldest

votes

up vote
3
down vote



accepted

The Gumbel distribution is not important here. In general, whenever $X$ and $Y$ have the same distribution and are independent, then $X-Y$ will be symmetric about zero. This is because $X-Y$ has the same joint distribution as $Y-X$, so for all $cin mathbb R$,
$$
P(X-Yle c)=P(Y-Xle -c)stackrelX-Ystackreld=Y-X=P(X-Yle -c)
$$
which says $X-Y$ is symmetric about $0$. This immediately implies its median of $X-Y$ is $0$, since $P(X-Yle 0)=P(X-Yge 0$) and these probabilities sum to $1+P(X-Y=0)ge 1$, so they must both be at least $frac12$.

share|cite|improve this answer

answered Jul 31 at 17:38

Mike Earnest

14.7k11644

add a comment | 

up vote
3
down vote



accepted

The Gumbel distribution is not important here. In general, whenever $X$ and $Y$ have the same distribution and are independent, then $X-Y$ will be symmetric about zero. This is because $X-Y$ has the same joint distribution as $Y-X$, so for all $cin mathbb R$,
$$
P(X-Yle c)=P(Y-Xle -c)stackrelX-Ystackreld=Y-X=P(X-Yle -c)
$$
which says $X-Y$ is symmetric about $0$. This immediately implies its median of $X-Y$ is $0$, since $P(X-Yle 0)=P(X-Yge 0$) and these probabilities sum to $1+P(X-Y=0)ge 1$, so they must both be at least $frac12$.

share|cite|improve this answer

answered Jul 31 at 17:38

Mike Earnest

14.7k11644

add a comment | 

up vote
3
down vote



accepted

up vote
3
down vote



accepted

The Gumbel distribution is not important here. In general, whenever $X$ and $Y$ have the same distribution and are independent, then $X-Y$ will be symmetric about zero. This is because $X-Y$ has the same joint distribution as $Y-X$, so for all $cin mathbb R$,
$$
P(X-Yle c)=P(Y-Xle -c)stackrelX-Ystackreld=Y-X=P(X-Yle -c)
$$
which says $X-Y$ is symmetric about $0$. This immediately implies its median of $X-Y$ is $0$, since $P(X-Yle 0)=P(X-Yge 0$) and these probabilities sum to $1+P(X-Y=0)ge 1$, so they must both be at least $frac12$.

share|cite|improve this answer

answered Jul 31 at 17:38

Mike Earnest

14.7k11644

The Gumbel distribution is not important here. In general, whenever $X$ and $Y$ have the same distribution and are independent, then $X-Y$ will be symmetric about zero. This is because $X-Y$ has the same joint distribution as $Y-X$, so for all $cin mathbb R$,
$$
P(X-Yle c)=P(Y-Xle -c)stackrelX-Ystackreld=Y-X=P(X-Yle -c)
$$
which says $X-Y$ is symmetric about $0$. This immediately implies its median of $X-Y$ is $0$, since $P(X-Yle 0)=P(X-Yge 0$) and these probabilities sum to $1+P(X-Y=0)ge 1$, so they must both be at least $frac12$.

share|cite|improve this answer

answered Jul 31 at 17:38

Mike Earnest

14.7k11644

share|cite|improve this answer

share|cite|improve this answer

answered Jul 31 at 17:38

Mike Earnest

14.7k11644

answered Jul 31 at 17:38

Mike Earnest

14.7k11644

answered Jul 31 at 17:38

Mike Earnest

14.7k11644

14.7k11644

add a comment | 

add a comment | 

up vote
2
down vote

Use this property: if $X$ and $Y$ are distributed as Gumbel distribution $textGum(alpha, beta)$ then $X - Y sim textLog(0, beta)$ (see Logistic distribution) and since for logistic distribution has median equal to location parameter $mu = 0$ then the $textmed V_j = 0$. Logistic distribution is symmetric as well, so the second part of your question is also true.

share|cite|improve this answer

answered Jul 31 at 15:24

pointguard0

689517

add a comment | 

up vote
2
down vote

Use this property: if $X$ and $Y$ are distributed as Gumbel distribution $textGum(alpha, beta)$ then $X - Y sim textLog(0, beta)$ (see Logistic distribution) and since for logistic distribution has median equal to location parameter $mu = 0$ then the $textmed V_j = 0$. Logistic distribution is symmetric as well, so the second part of your question is also true.

share|cite|improve this answer

answered Jul 31 at 15:24

pointguard0

689517

add a comment | 

up vote
2
down vote

up vote
2
down vote

Use this property: if $X$ and $Y$ are distributed as Gumbel distribution $textGum(alpha, beta)$ then $X - Y sim textLog(0, beta)$ (see Logistic distribution) and since for logistic distribution has median equal to location parameter $mu = 0$ then the $textmed V_j = 0$. Logistic distribution is symmetric as well, so the second part of your question is also true.

share|cite|improve this answer

answered Jul 31 at 15:24

pointguard0

689517

Use this property: if $X$ and $Y$ are distributed as Gumbel distribution $textGum(alpha, beta)$ then $X - Y sim textLog(0, beta)$ (see Logistic distribution) and since for logistic distribution has median equal to location parameter $mu = 0$ then the $textmed V_j = 0$. Logistic distribution is symmetric as well, so the second part of your question is also true.

share|cite|improve this answer

answered Jul 31 at 15:24

pointguard0

689517

share|cite|improve this answer

share|cite|improve this answer

answered Jul 31 at 15:24

pointguard0

689517

answered Jul 31 at 15:24

pointguard0

689517

answered Jul 31 at 15:24

pointguard0

689517

689517

add a comment | 

add a comment |