finding correlation coefficient given conditional expectations

Question

Given $Y1$ and $Y2$ have a bivariate normal distribution where $E(Y1|Y2)=4.7-0.16Y2$ and $E(Y2|Y1)=0.8-Y1$ and conditional variance is 3.64. How can I find the correlation. I have tried the $E(E(Y1|Y2))=E(Y1)=4.7-0.16E(Y2)$ but it doesnt seem to help get the $pdf$ of $Y1$ ar $Y2$.

Answer 1

Marshall your facts: for the bivariate normal, $E[Y_1|Y_2]=\mu_1+{\rho\sigma_1\over\sigma_2}(Y_2-\mu_2)$ (where $\mu_i$ is the mean of $Y_i$, $\sigma_i$ is its standard deviation, and $\rho$ is the correlation of $Y_1$ and $Y_2$.) From this you can deduce that ${\rho\sigma_1\over\sigma_2}=-0.16$. Likewise (reverse the roles of $Y_1$ and $Y_2$), ${\rho\sigma_2\over\sigma_1} = -1$. You now know $\rho^2$.

I presume the "conditional variance" you refer to is that of $Y_2$ given $Y_1$; in general this conditional variance is $(1-\rho^2)\sigma_2^2$. You can now deduce $\sigma_2$, and then $\sigma_1$. Knowing these it is an easy calculation to find $\mu_1$ and $\mu_2$ from the given information.