Finding joint pdf of two functions, one min of two geometric functions and other defined based on difference...
0
$begingroup$
I'm really stuck on this problem and I have no clue how to proceed right now.
Let $X,Y$ be random geometric variables. Let $Z = min(X,Y)$, and $W = {
left{
begin{array}{lll}
0 & mbox{if } X<Y \
1 & mbox{if } X=Y \
2 & mbox{if } X>Y
end{array}
right.
}$
Find the joint pdf of $Z$ and $W$.
I'm not really sure where to go with this problem. Any suggestions for where to start would be greatly appreciated.
probability probability-distributions
asked Jan 23 at 0:04
rocketPoweredrocketPowered
244
$endgroup$
add a comment |
0
$begingroup$
I'm really stuck on this problem and I have no clue how to proceed right now.
Let $X,Y$ be random geometric variables. Let $Z = min(X,Y)$, and $W = {
left{
begin{array}{lll}
0 & mbox{if } X<Y \
1 & mbox{if } X=Y \
2 & mbox{if } X>Y
end{array}
right.
}$
Find the joint pdf of $Z$ and $W$.
I'm not really sure where to go with this problem. Any suggestions for where to start would be greatly appreciated.
probability probability-distributions
asked Jan 23 at 0:04
rocketPoweredrocketPowered
244
$endgroup$
$begingroup$
First you think about the values that $(Z,W)$ take on. For instance does when$X<Y$? what is $(Z,W)$? etc. Then compute the probabilities of those values. By the way no density at play here since you are working with discrete RVs, its all PMFs and a joint PMF.
$endgroup$
– LoveTooNap29
Jan 23 at 1:03
$begingroup$
It could also be useful to know how the minimum of two (presumably identically and independently distributed) geometric RVs is distributed... maybe start there too. Its been asked on this site many times.
$endgroup$
– LoveTooNap29
Jan 23 at 1:05
$begingroup$
@LoveTooNap29 I can see that when $X < Y, (Z,W) = (X, 0)$, but I'm not quite sure how to turn this into a joint pmf. I'm trying to do this without the assumption that $Z$ and $W$ are independent, so I don't want to just multiply the pmf of $X$ by the probability that $X < Y$. How would I go about finding the pmf for $(Z,W) = (X,0)$ without using independence?
$endgroup$
– rocketPowered
Jan 23 at 2:10
add a comment |
0
0
0
$begingroup$
I'm really stuck on this problem and I have no clue how to proceed right now.
Let $X,Y$ be random geometric variables. Let $Z = min(X,Y)$, and $W = {
left{
begin{array}{lll}
0 & mbox{if } X<Y \
1 & mbox{if } X=Y \
2 & mbox{if } X>Y
end{array}
right.
}$
Find the joint pdf of $Z$ and $W$.
I'm not really sure where to go with this problem. Any suggestions for where to start would be greatly appreciated.
probability probability-distributions
asked Jan 23 at 0:04
rocketPoweredrocketPowered
244
$endgroup$
I'm really stuck on this problem and I have no clue how to proceed right now.
Let $X,Y$ be random geometric variables. Let $Z = min(X,Y)$, and $W = {
left{
begin{array}{lll}
0 & mbox{if } X<Y \
1 & mbox{if } X=Y \
2 & mbox{if } X>Y
end{array}
right.
}$
Find the joint pdf of $Z$ and $W$.
I'm not really sure where to go with this problem. Any suggestions for where to start would be greatly appreciated.
probability probability-distributions
probability probability-distributions
asked Jan 23 at 0:04
rocketPoweredrocketPowered
244
asked Jan 23 at 0:04
rocketPoweredrocketPowered
244
asked Jan 23 at 0:04
rocketPoweredrocketPowered
244
asked Jan 23 at 0:04
rocketPoweredrocketPowered
244
asked Jan 23 at 0:04
rocketPoweredrocketPowered
244
244
$begingroup$
First you think about the values that $(Z,W)$ take on. For instance does when$X<Y$? what is $(Z,W)$? etc. Then compute the probabilities of those values. By the way no density at play here since you are working with discrete RVs, its all PMFs and a joint PMF.
$endgroup$
– LoveTooNap29
Jan 23 at 1:03
$begingroup$
It could also be useful to know how the minimum of two (presumably identically and independently distributed) geometric RVs is distributed... maybe start there too. Its been asked on this site many times.
$endgroup$
– LoveTooNap29
Jan 23 at 1:05
$begingroup$
@LoveTooNap29 I can see that when $X < Y, (Z,W) = (X, 0)$, but I'm not quite sure how to turn this into a joint pmf. I'm trying to do this without the assumption that $Z$ and $W$ are independent, so I don't want to just multiply the pmf of $X$ by the probability that $X < Y$. How would I go about finding the pmf for $(Z,W) = (X,0)$ without using independence?
$endgroup$
– rocketPowered
Jan 23 at 2:10
add a comment |
$begingroup$
First you think about the values that $(Z,W)$ take on. For instance does when$X<Y$? what is $(Z,W)$? etc. Then compute the probabilities of those values. By the way no density at play here since you are working with discrete RVs, its all PMFs and a joint PMF.
$endgroup$
– LoveTooNap29
Jan 23 at 1:03
$begingroup$
It could also be useful to know how the minimum of two (presumably identically and independently distributed) geometric RVs is distributed... maybe start there too. Its been asked on this site many times.
$endgroup$
– LoveTooNap29
Jan 23 at 1:05
$begingroup$
@LoveTooNap29 I can see that when $X < Y, (Z,W) = (X, 0)$, but I'm not quite sure how to turn this into a joint pmf. I'm trying to do this without the assumption that $Z$ and $W$ are independent, so I don't want to just multiply the pmf of $X$ by the probability that $X < Y$. How would I go about finding the pmf for $(Z,W) = (X,0)$ without using independence?
$endgroup$
– rocketPowered
Jan 23 at 2:10
$begingroup$
First you think about the values that $(Z,W)$ take on. For instance does when$X<Y$? what is $(Z,W)$? etc. Then compute the probabilities of those values. By the way no density at play here since you are working with discrete RVs, its all PMFs and a joint PMF.
$endgroup$
– LoveTooNap29
Jan 23 at 1:03
$begingroup$
First you think about the values that $(Z,W)$ take on. For instance does when$X<Y$? what is $(Z,W)$? etc. Then compute the probabilities of those values. By the way no density at play here since you are working with discrete RVs, its all PMFs and a joint PMF.
$endgroup$
– LoveTooNap29
Jan 23 at 1:03
$begingroup$
It could also be useful to know how the minimum of two (presumably identically and independently distributed) geometric RVs is distributed... maybe start there too. Its been asked on this site many times.
$endgroup$
– LoveTooNap29
Jan 23 at 1:05
$begingroup$
It could also be useful to know how the minimum of two (presumably identically and independently distributed) geometric RVs is distributed... maybe start there too. Its been asked on this site many times.
$endgroup$
– LoveTooNap29
Jan 23 at 1:05
$begingroup$
@LoveTooNap29 I can see that when $X < Y, (Z,W) = (X, 0)$, but I'm not quite sure how to turn this into a joint pmf. I'm trying to do this without the assumption that $Z$ and $W$ are independent, so I don't want to just multiply the pmf of $X$ by the probability that $X < Y$. How would I go about finding the pmf for $(Z,W) = (X,0)$ without using independence?
$endgroup$
– rocketPowered
Jan 23 at 2:10
$begingroup$
@LoveTooNap29 I can see that when $X < Y, (Z,W) = (X, 0)$, but I'm not quite sure how to turn this into a joint pmf. I'm trying to do this without the assumption that $Z$ and $W$ are independent, so I don't want to just multiply the pmf of $X$ by the probability that $X < Y$. How would I go about finding the pmf for $(Z,W) = (X,0)$ without using independence?
$endgroup$
– rocketPowered
Jan 23 at 2:10
add a comment |
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$begingroup$
First you think about the values that $(Z,W)$ take on. For instance does when$X<Y$? what is $(Z,W)$? etc. Then compute the probabilities of those values. By the way no density at play here since you are working with discrete RVs, its all PMFs and a joint PMF.
$endgroup$
– LoveTooNap29
Jan 23 at 1:03
$begingroup$
It could also be useful to know how the minimum of two (presumably identically and independently distributed) geometric RVs is distributed... maybe start there too. Its been asked on this site many times.
$endgroup$
– LoveTooNap29
Jan 23 at 1:05
$begingroup$
@LoveTooNap29 I can see that when $X < Y, (Z,W) = (X, 0)$, but I'm not quite sure how to turn this into a joint pmf. I'm trying to do this without the assumption that $Z$ and $W$ are independent, so I don't want to just multiply the pmf of $X$ by the probability that $X < Y$. How would I go about finding the pmf for $(Z,W) = (X,0)$ without using independence?
$endgroup$
– rocketPowered
Jan 23 at 2:10