Intuition for this characterization of the exponential law
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Let $X$ and $Y$ be two real independant random variables having same distribution $mu$ with density $f$ which is positive only on $mathbb R ^+$. Let $U = min(X,Y)$ and $V = |X-Y|$. The result I was able to prove in a guided exercise is the following :
Assume that $f$ is continuous on $mathbb R ^+$. Then $U$ and $V$ are independant if and only if $mu$ is an exponential law.
I would like to have some intuition as to how to interpret this result. The only intuitive way I can think of the exponential law is that it is the only law with "no memory". How does this relate to the facts that
- for the exponential law, $U$ and $V$ are independant ?
- the exponential law is the only such law ?
Any explanation about this phenomenon are welcome.
probability probability-distributions
asked Nov 27 at 13:30
Suzet
2,574527
add a comment |
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0
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Let $X$ and $Y$ be two real independant random variables having same distribution $mu$ with density $f$ which is positive only on $mathbb R ^+$. Let $U = min(X,Y)$ and $V = |X-Y|$. The result I was able to prove in a guided exercise is the following :
Assume that $f$ is continuous on $mathbb R ^+$. Then $U$ and $V$ are independant if and only if $mu$ is an exponential law.
I would like to have some intuition as to how to interpret this result. The only intuitive way I can think of the exponential law is that it is the only law with "no memory". How does this relate to the facts that
- for the exponential law, $U$ and $V$ are independant ?
- the exponential law is the only such law ?
Any explanation about this phenomenon are welcome.
probability probability-distributions
asked Nov 27 at 13:30
Suzet
2,574527
One part of the proof previously asked here. Regarding the characterization, you might have a look at this paper by T.S. Ferguson.
– StubbornAtom
Dec 1 at 11:17
add a comment |
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0
down vote
favorite
up vote
0
down vote
favorite
Let $X$ and $Y$ be two real independant random variables having same distribution $mu$ with density $f$ which is positive only on $mathbb R ^+$. Let $U = min(X,Y)$ and $V = |X-Y|$. The result I was able to prove in a guided exercise is the following :
Assume that $f$ is continuous on $mathbb R ^+$. Then $U$ and $V$ are independant if and only if $mu$ is an exponential law.
I would like to have some intuition as to how to interpret this result. The only intuitive way I can think of the exponential law is that it is the only law with "no memory". How does this relate to the facts that
- for the exponential law, $U$ and $V$ are independant ?
- the exponential law is the only such law ?
Any explanation about this phenomenon are welcome.
probability probability-distributions
asked Nov 27 at 13:30
Suzet
2,574527
Let $X$ and $Y$ be two real independant random variables having same distribution $mu$ with density $f$ which is positive only on $mathbb R ^+$. Let $U = min(X,Y)$ and $V = |X-Y|$. The result I was able to prove in a guided exercise is the following :
Assume that $f$ is continuous on $mathbb R ^+$. Then $U$ and $V$ are independant if and only if $mu$ is an exponential law.
I would like to have some intuition as to how to interpret this result. The only intuitive way I can think of the exponential law is that it is the only law with "no memory". How does this relate to the facts that
- for the exponential law, $U$ and $V$ are independant ?
- the exponential law is the only such law ?
Any explanation about this phenomenon are welcome.
probability probability-distributions
probability probability-distributions
asked Nov 27 at 13:30
Suzet
2,574527
asked Nov 27 at 13:30
Suzet
2,574527
asked Nov 27 at 13:30
Suzet
2,574527
asked Nov 27 at 13:30
Suzet
2,574527
asked Nov 27 at 13:30
Suzet
2,574527
2,574527
One part of the proof previously asked here. Regarding the characterization, you might have a look at this paper by T.S. Ferguson.
– StubbornAtom
Dec 1 at 11:17
add a comment |
One part of the proof previously asked here. Regarding the characterization, you might have a look at this paper by T.S. Ferguson.
– StubbornAtom
Dec 1 at 11:17
One part of the proof previously asked here. Regarding the characterization, you might have a look at this paper by T.S. Ferguson.
– StubbornAtom
Dec 1 at 11:17
One part of the proof previously asked here. Regarding the characterization, you might have a look at this paper by T.S. Ferguson.
– StubbornAtom
Dec 1 at 11:17
add a comment |
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One part of the proof previously asked here. Regarding the characterization, you might have a look at this paper by T.S. Ferguson.
– StubbornAtom
Dec 1 at 11:17