Expectation computation of correlated normal random variables
$begingroup$
I'm struggling to prove the following maybe some of you can help me. Here is my problem
Let $mathbf{X}simmathcal{N}(mu,Sigma)$ where $mathbf{X}=pmatrix{X_1\X_2}, mu=pmatrix{mu_1\mu_2}$ and $Sigma=pmatrix{sigma_1^2&rhosigma_1sigma_2\ rhosigma_1sigma_2&sigma_2^2}$. Show that $$mathbb{E}[g(X_1)exp(-X_2)]=mathbb{E}[g(X_1-rhosigma_1sigma_2)]mathbb{E}[exp(-X_2)]$$
where $g$ is a function for which the above expectations exist.
I tried to make several attempts like playing with the conditional expectations but I didn't succeed to prove anything. I also tried to compute the double integrals of the right term and the left term and make some change of variables but same, I didn't manage to find anything.
I hope someone can give me some hints.
Thanks in advance!
Cheers & merry Christmas!
normal-distribution correlation expected-value bivariate-distributions
asked Dec 24 '18 at 14:21
user2076694user2076694
1313
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add a comment |
$begingroup$
I'm struggling to prove the following maybe some of you can help me. Here is my problem
Let $mathbf{X}simmathcal{N}(mu,Sigma)$ where $mathbf{X}=pmatrix{X_1\X_2}, mu=pmatrix{mu_1\mu_2}$ and $Sigma=pmatrix{sigma_1^2&rhosigma_1sigma_2\ rhosigma_1sigma_2&sigma_2^2}$. Show that $$mathbb{E}[g(X_1)exp(-X_2)]=mathbb{E}[g(X_1-rhosigma_1sigma_2)]mathbb{E}[exp(-X_2)]$$
where $g$ is a function for which the above expectations exist.
I tried to make several attempts like playing with the conditional expectations but I didn't succeed to prove anything. I also tried to compute the double integrals of the right term and the left term and make some change of variables but same, I didn't manage to find anything.
I hope someone can give me some hints.
Thanks in advance!
Cheers & merry Christmas!
normal-distribution correlation expected-value bivariate-distributions
asked Dec 24 '18 at 14:21
user2076694user2076694
1313
$endgroup$
add a comment |
$begingroup$
I'm struggling to prove the following maybe some of you can help me. Here is my problem
Let $mathbf{X}simmathcal{N}(mu,Sigma)$ where $mathbf{X}=pmatrix{X_1\X_2}, mu=pmatrix{mu_1\mu_2}$ and $Sigma=pmatrix{sigma_1^2&rhosigma_1sigma_2\ rhosigma_1sigma_2&sigma_2^2}$. Show that $$mathbb{E}[g(X_1)exp(-X_2)]=mathbb{E}[g(X_1-rhosigma_1sigma_2)]mathbb{E}[exp(-X_2)]$$
where $g$ is a function for which the above expectations exist.
I tried to make several attempts like playing with the conditional expectations but I didn't succeed to prove anything. I also tried to compute the double integrals of the right term and the left term and make some change of variables but same, I didn't manage to find anything.
I hope someone can give me some hints.
Thanks in advance!
Cheers & merry Christmas!
normal-distribution correlation expected-value bivariate-distributions
asked Dec 24 '18 at 14:21
user2076694user2076694
1313
$endgroup$
I'm struggling to prove the following maybe some of you can help me. Here is my problem
Let $mathbf{X}simmathcal{N}(mu,Sigma)$ where $mathbf{X}=pmatrix{X_1\X_2}, mu=pmatrix{mu_1\mu_2}$ and $Sigma=pmatrix{sigma_1^2&rhosigma_1sigma_2\ rhosigma_1sigma_2&sigma_2^2}$. Show that $$mathbb{E}[g(X_1)exp(-X_2)]=mathbb{E}[g(X_1-rhosigma_1sigma_2)]mathbb{E}[exp(-X_2)]$$
where $g$ is a function for which the above expectations exist.
I tried to make several attempts like playing with the conditional expectations but I didn't succeed to prove anything. I also tried to compute the double integrals of the right term and the left term and make some change of variables but same, I didn't manage to find anything.
I hope someone can give me some hints.
Thanks in advance!
Cheers & merry Christmas!
normal-distribution correlation expected-value bivariate-distributions
normal-distribution correlation expected-value bivariate-distributions
asked Dec 24 '18 at 14:21
user2076694user2076694
1313
asked Dec 24 '18 at 14:21
user2076694user2076694
1313
asked Dec 24 '18 at 14:21
user2076694user2076694
1313
asked Dec 24 '18 at 14:21
user2076694user2076694
1313
asked Dec 24 '18 at 14:21
user2076694user2076694
1313
1313
add a comment |
add a comment |
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