Mixing
Joint distribution of min and absolute difference
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Let $X_1, X_2 sim Exp(1)$ be independent random variables. Let $Y = min{X_1,X_2}, Z = |X_1-X_2|$. What is joint distribution of $Y,Z$?
I think I can figure out the individual distributions of $Y$ and of $Z$. e.g. for $Y$,
$$P(Y leq x) = P(X_1 leq x lor X_2 leq x) = 1 - P(X_1 > x land X_2 > x) = 1 - P(X_1 > x)P(X_2 > x)$$
and for $Z$ I can break down $|X_1 - X_2|$ into the two cases $X_1 > X_2$, $X_2 > X_1$. But when it comes to the joint distribution I don't know where to start, because there seems to be dependence between $Y,Z$. E.g. if $Y$ is large, then both $X_1, X_2$ are large, which in my mind would make the probability of a large $Z$ value small.
I suspect I may have to use the memorylessness property of the exponential distribution but I don't know how.
probability probability-theory probability-distributions
asked Feb 3 at 12:39
D GD G
1779
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add a comment |
$begingroup$
Let $X_1, X_2 sim Exp(1)$ be independent random variables. Let $Y = min{X_1,X_2}, Z = |X_1-X_2|$. What is joint distribution of $Y,Z$?
I think I can figure out the individual distributions of $Y$ and of $Z$. e.g. for $Y$,
$$P(Y leq x) = P(X_1 leq x lor X_2 leq x) = 1 - P(X_1 > x land X_2 > x) = 1 - P(X_1 > x)P(X_2 > x)$$
and for $Z$ I can break down $|X_1 - X_2|$ into the two cases $X_1 > X_2$, $X_2 > X_1$. But when it comes to the joint distribution I don't know where to start, because there seems to be dependence between $Y,Z$. E.g. if $Y$ is large, then both $X_1, X_2$ are large, which in my mind would make the probability of a large $Z$ value small.
I suspect I may have to use the memorylessness property of the exponential distribution but I don't know how.
probability probability-theory probability-distributions
asked Feb 3 at 12:39
D GD G
1779
$endgroup$
$begingroup$
See math.stackexchange.com/questions/2240822/… for why $Y$ and $Z$ are independently distributed.
$endgroup$
– StubbornAtom
Feb 3 at 16:43
add a comment |
$begingroup$
Let $X_1, X_2 sim Exp(1)$ be independent random variables. Let $Y = min{X_1,X_2}, Z = |X_1-X_2|$. What is joint distribution of $Y,Z$?
I think I can figure out the individual distributions of $Y$ and of $Z$. e.g. for $Y$,
$$P(Y leq x) = P(X_1 leq x lor X_2 leq x) = 1 - P(X_1 > x land X_2 > x) = 1 - P(X_1 > x)P(X_2 > x)$$
and for $Z$ I can break down $|X_1 - X_2|$ into the two cases $X_1 > X_2$, $X_2 > X_1$. But when it comes to the joint distribution I don't know where to start, because there seems to be dependence between $Y,Z$. E.g. if $Y$ is large, then both $X_1, X_2$ are large, which in my mind would make the probability of a large $Z$ value small.
I suspect I may have to use the memorylessness property of the exponential distribution but I don't know how.
probability probability-theory probability-distributions
asked Feb 3 at 12:39
D GD G
1779
$endgroup$
Let $X_1, X_2 sim Exp(1)$ be independent random variables. Let $Y = min{X_1,X_2}, Z = |X_1-X_2|$. What is joint distribution of $Y,Z$?
I think I can figure out the individual distributions of $Y$ and of $Z$. e.g. for $Y$,
$$P(Y leq x) = P(X_1 leq x lor X_2 leq x) = 1 - P(X_1 > x land X_2 > x) = 1 - P(X_1 > x)P(X_2 > x)$$
and for $Z$ I can break down $|X_1 - X_2|$ into the two cases $X_1 > X_2$, $X_2 > X_1$. But when it comes to the joint distribution I don't know where to start, because there seems to be dependence between $Y,Z$. E.g. if $Y$ is large, then both $X_1, X_2$ are large, which in my mind would make the probability of a large $Z$ value small.
I suspect I may have to use the memorylessness property of the exponential distribution but I don't know how.
probability probability-theory probability-distributions
probability probability-theory probability-distributions
asked Feb 3 at 12:39
D GD G
1779
asked Feb 3 at 12:39
D GD G
1779
asked Feb 3 at 12:39
D GD G
1779
asked Feb 3 at 12:39
D GD G
1779
asked Feb 3 at 12:39
D GD G
1779
1779
$begingroup$
See math.stackexchange.com/questions/2240822/… for why $Y$ and $Z$ are independently distributed.
$endgroup$
– StubbornAtom
Feb 3 at 16:43
add a comment |
$begingroup$
See math.stackexchange.com/questions/2240822/… for why $Y$ and $Z$ are independently distributed.
$endgroup$
– StubbornAtom
Feb 3 at 16:43
$begingroup$
See math.stackexchange.com/questions/2240822/… for why $Y$ and $Z$ are independently distributed.
$endgroup$
– StubbornAtom
Feb 3 at 16:43
$begingroup$
See math.stackexchange.com/questions/2240822/… for why $Y$ and $Z$ are independently distributed.
$endgroup$
– StubbornAtom
Feb 3 at 16:43
add a comment |
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$begingroup$
See math.stackexchange.com/questions/2240822/… for why $Y$ and $Z$ are independently distributed.
$endgroup$
– StubbornAtom
Feb 3 at 16:43