Correlation of function of two random variables after resampling one
$begingroup$
Consider a measurable function $g:mathbb{R}^2rightarrow [0,+infty)^2$ that satisfies that for all $(x,y)inmathbb{R}^2$ at least one coordinate
of $g(x,y)$ is 0. Call its coordinates $g_1$ and $g_2$.
Let $X,Y_1, Y_2$ be independent real-valued random variables, such that $Y_1$ and $Y_2$ have the same distribution, and such that $mathbb{E}(g_2(X,Y_1))<infty$.
Is the following inequality true?
$ mathbb{E}Big(g_2(X,Y_2)cdot mathbb{1}_{{g_1(X,Y_1)=0}} Big)
geq mathbb{E}Big(g_2(X,Y_2)Big) cdot mathbb{P}Big(g_1(X,Y_1)=0Big)$
It looks simple and intuitive, but so far I haven't found a proof or a counterexample.
probability random-variables correlation expected-value
asked Dec 14 '18 at 19:54
Andrés CristiAndrés Cristi
11
$endgroup$
add a comment |
$begingroup$
Consider a measurable function $g:mathbb{R}^2rightarrow [0,+infty)^2$ that satisfies that for all $(x,y)inmathbb{R}^2$ at least one coordinate
of $g(x,y)$ is 0. Call its coordinates $g_1$ and $g_2$.
Let $X,Y_1, Y_2$ be independent real-valued random variables, such that $Y_1$ and $Y_2$ have the same distribution, and such that $mathbb{E}(g_2(X,Y_1))<infty$.
Is the following inequality true?
$ mathbb{E}Big(g_2(X,Y_2)cdot mathbb{1}_{{g_1(X,Y_1)=0}} Big)
geq mathbb{E}Big(g_2(X,Y_2)Big) cdot mathbb{P}Big(g_1(X,Y_1)=0Big)$
It looks simple and intuitive, but so far I haven't found a proof or a counterexample.
probability random-variables correlation expected-value
asked Dec 14 '18 at 19:54
Andrés CristiAndrés Cristi
11
$endgroup$
add a comment |
$begingroup$
Consider a measurable function $g:mathbb{R}^2rightarrow [0,+infty)^2$ that satisfies that for all $(x,y)inmathbb{R}^2$ at least one coordinate
of $g(x,y)$ is 0. Call its coordinates $g_1$ and $g_2$.
Let $X,Y_1, Y_2$ be independent real-valued random variables, such that $Y_1$ and $Y_2$ have the same distribution, and such that $mathbb{E}(g_2(X,Y_1))<infty$.
Is the following inequality true?
$ mathbb{E}Big(g_2(X,Y_2)cdot mathbb{1}_{{g_1(X,Y_1)=0}} Big)
geq mathbb{E}Big(g_2(X,Y_2)Big) cdot mathbb{P}Big(g_1(X,Y_1)=0Big)$
It looks simple and intuitive, but so far I haven't found a proof or a counterexample.
probability random-variables correlation expected-value
asked Dec 14 '18 at 19:54
Andrés CristiAndrés Cristi
11
$endgroup$
Consider a measurable function $g:mathbb{R}^2rightarrow [0,+infty)^2$ that satisfies that for all $(x,y)inmathbb{R}^2$ at least one coordinate
of $g(x,y)$ is 0. Call its coordinates $g_1$ and $g_2$.
Let $X,Y_1, Y_2$ be independent real-valued random variables, such that $Y_1$ and $Y_2$ have the same distribution, and such that $mathbb{E}(g_2(X,Y_1))<infty$.
Is the following inequality true?
$ mathbb{E}Big(g_2(X,Y_2)cdot mathbb{1}_{{g_1(X,Y_1)=0}} Big)
geq mathbb{E}Big(g_2(X,Y_2)Big) cdot mathbb{P}Big(g_1(X,Y_1)=0Big)$
It looks simple and intuitive, but so far I haven't found a proof or a counterexample.
probability random-variables correlation expected-value
probability random-variables correlation expected-value
asked Dec 14 '18 at 19:54
Andrés CristiAndrés Cristi
11
asked Dec 14 '18 at 19:54
Andrés CristiAndrés Cristi
11
asked Dec 14 '18 at 19:54
Andrés CristiAndrés Cristi
11
asked Dec 14 '18 at 19:54
Andrés CristiAndrés Cristi
11
asked Dec 14 '18 at 19:54
Andrés CristiAndrés Cristi
11
11
add a comment |
add a comment |
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