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Suppose that X and Y are random variables such that E(Y | X) = 7 - (1/4)x and E(X | Y) = 10 - Y . Determine the correlation of X and Y .


So far I've got

E(x)=4 E(y)=6

Now I'm trying to find

E(xy) to use in cov(x,y)=E(xy)-E(x)E(y)



all to use in cor(x,y)=cov(x,y)/(v(x)v(y))^.5

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Correlation as measured by...? –  Xodarap Dec 5 '12 at 3:15
cor(x,y) = cov(x,y) / (v(x)v(y))^.5 ...Im not sure I understand your question though –  Dan Dec 5 '12 at 3:20
Pearson's is a fine way to define correlation; it's just not the only way. –  Xodarap Dec 5 '12 at 3:22

1 Answer 1

Hint: $E[Y\mid X]$ is the minimum-mean-square-error estimator of $Y$ given the value of $X$. The linear minimum-mean-square-error estimator of $Y$ given the value of $X$ is

$$\hat{Y} = \mu_Y + \frac{\rho\sigma_Y}{\sigma_X}(X-\mu_X).$$

Similar statements apply to $E[X\mid Y]$ etc. Just interchange $X$ and $Y$ in the above formulas.

Now, if $E[Y\mid X]$ is a linear function of $X$ and $E[X\mid Y]$ is a linear function of $Y$, can you use the known forms of the linear minimum-mean-square-error estimators to deduce the value of $\rho$?

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yeah Ive found E(y)=6. Any other hints? –  Dan Dec 5 '12 at 3:28
You don't need to find $E[Y]$ or $E[X]$. Hint: show that the coefficients of $X$ and $Y$ in the equations given to you are $\rho\sigma_Y/\sigma_X$ and $\rho\sigma_X/\sigma_Y$ and use the information given to you to deduce that $\rho=0.5$ without needing to find $E[X]$ or $E[Y]$ or $\text{var}(X)$ or $\text{var}(Y)$ or $\text{cov}(X,Y)$ or $E[XY]$. –  Dilip Sarwate Dec 5 '12 at 3:35
@DilipSarwate (+1), but I think you meant to write $\rho = -1/2$ in your comment. –  r.e.s. Dec 5 '12 at 4:16
@r.e.s. Yes, you are correct. I realized that I had neglected to take the sign of $\rho$ into account after it was too late to edit the comment and was going to write a further comment. Thanks for the upvote. –  Dilip Sarwate Dec 5 '12 at 11:25

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