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KevinKim
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Construct joint distribution of $X,Y$ such that $E[X|Y=y]$$E[X|Y=y,y\geq \bar{y}]$ is piecewise linear

Here are three questions all related to uniform distribution:

Can one construct a joint density $f(x,y)$ such that the marginal distribution is $U[a,b]$ for $X$, $U[c,d]$ for $Y$ and the correlation between $X$ and $Y$ is $\rho\in(0,1)$? Is there a simple mathematical expression for $f(x,y)$?

Can one construct a joint density $f(x,y)$ such that the marginal distribution of $Y\sim~U[c,d]$, no restrictions on $X$ as long as it has finite support such that $\mathbb{E}[X|Y=y]$ is a piecewise linear function of $y$?

The third one is: Can one construct a joint density $f(x,y)$ such(it would be great that the marginal distribution of $Y\sim~U[c,d]$, no restrictions on $X$ also has uniform distribution) as long as it has finite support such that $\mathbb{E}[X|Y=y, Y\geq \bar{y}]$ (where $\bar{y}$ is some pre-specified number that is within the support of $Y$) is a piecewise linear function of $y$?

Here is some background of this problem. $X$ is the true state of a system and $Y$ is some index of the state which is not perfect but it is related to $X$. An agent observe $Y$ and want to learn the true state $X$. But the agent was told that $Y$ is greater than some threshold $\bar{y}$, i.e., the agent know that he cannot observe $Y$ that is strictly lower than $\bar{y}$, a pre-specified constant. I've read this (Is there a parametric joint distribution such that $X$ and $Y$ are both uniform and $\mathbb{E}[Y \;|\; X]$ is linear?). It is relevant to my problem but I think they are not completely the same.

Construct joint distribution of $X,Y$ such that $E[X|Y=y]$ is piecewise linear

Here are three questions all related to uniform distribution:

Can one construct a joint density $f(x,y)$ such that the marginal distribution is $U[a,b]$ for $X$, $U[c,d]$ for $Y$ and the correlation between $X$ and $Y$ is $\rho\in(0,1)$? Is there a simple mathematical expression for $f(x,y)$?

Can one construct a joint density $f(x,y)$ such that the marginal distribution of $Y\sim~U[c,d]$, no restrictions on $X$ as long as it has finite support such that $\mathbb{E}[X|Y=y]$ is a piecewise linear function of $y$?

The third one is: Can one construct a joint density $f(x,y)$ such that the marginal distribution of $Y\sim~U[c,d]$, no restrictions on $X$ as long as it has finite support such that $\mathbb{E}[X|Y=y, Y\geq \bar{y}]$ (where $\bar{y}$ is some pre-specified number that is within the support of $Y$) is a piecewise linear function of $y$?

Construct joint distribution of $X,Y$ such that $E[X|Y=y,y\geq \bar{y}]$ is piecewise linear

Can one construct a joint density $f(x,y)$ such that the marginal distribution of $Y\sim~U[c,d]$, no restrictions on $X$ (it would be great that $X$ also has uniform distribution) as long as it has finite support such that $\mathbb{E}[X|Y=y, Y\geq \bar{y}]$ (where $\bar{y}$ is some pre-specified number that is within the support of $Y$) is a piecewise linear function of $y$?

Here is some background of this problem. $X$ is the true state of a system and $Y$ is some index of the state which is not perfect but it is related to $X$. An agent observe $Y$ and want to learn the true state $X$. But the agent was told that $Y$ is greater than some threshold $\bar{y}$, i.e., the agent know that he cannot observe $Y$ that is strictly lower than $\bar{y}$, a pre-specified constant. I've read this (Is there a parametric joint distribution such that $X$ and $Y$ are both uniform and $\mathbb{E}[Y \;|\; X]$ is linear?). It is relevant to my problem but I think they are not completely the same.

Source Link
KevinKim
  • 6.9k
  • 4
  • 27
  • 36

Construct joint distribution of $X,Y$ such that $E[X|Y=y]$ is piecewise linear

Here are three questions all related to uniform distribution:

Can one construct a joint density $f(x,y)$ such that the marginal distribution is $U[a,b]$ for $X$, $U[c,d]$ for $Y$ and the correlation between $X$ and $Y$ is $\rho\in(0,1)$? Is there a simple mathematical expression for $f(x,y)$?

Can one construct a joint density $f(x,y)$ such that the marginal distribution of $Y\sim~U[c,d]$, no restrictions on $X$ as long as it has finite support such that $\mathbb{E}[X|Y=y]$ is a piecewise linear function of $y$?

The third one is: Can one construct a joint density $f(x,y)$ such that the marginal distribution of $Y\sim~U[c,d]$, no restrictions on $X$ as long as it has finite support such that $\mathbb{E}[X|Y=y, Y\geq \bar{y}]$ (where $\bar{y}$ is some pre-specified number that is within the support of $Y$) is a piecewise linear function of $y$?