Correlation coefficient and Expectation of two dimensional normal distribution.

Question

Random variable (X,Y) is normally distributed.

Conditional expectations are

$E(X|Y=y)=0.25y + 2$

$E(Y|X=x)=x-2$

How can i determine correlation coefficient and when that is known, the expectations of X and Y ?

Answer 1

If $ X \choose y$ is normally distributed, then $P(X|Y=y)=\mu _x+\rho \frac{\sigma _x}{\sigma _y}(Y-\mu _y) \quad (1)$

Or other way round.

If $ X \choose y$ is normally distributed, then $P(Y|X=x)=\mu _y+\rho \frac{\sigma _y}{\sigma _x}(X-\mu _x) \quad (2)$

You can substitue: $\mu _x=a,\mu _y=b,\frac{\sigma _x}{\sigma _y}=c$. After you have transformed the terms above, you can compare the coefficients and write down four equations. With this equation system the values of $\rho$ can be determinated.

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Transforming (1)

$\underbrace{\mu _x-\rho\frac{\sigma _x}{\sigma _y}\mu_y}_{2}+\underbrace{\rho \frac{\sigma _x}{\sigma _y}}_{0.25}y$

$\underbrace{a-\rho \cdot c \cdot b}_{2}+\underbrace{\rho \cdot c}_{0.25}\cdot y$

Thus $a-\rho \cdot c \cdot b=2$ and $\rho \cdot c=0.25 \quad (1a)$

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Transforming (2)

$\underbrace{\mu _y-\rho\frac{\sigma _y}{\sigma _x}\mu_x}_{-2}+\underbrace{\rho \frac{\sigma _y}{\sigma _x}}_{1}x$

$\underbrace{b-\rho \cdot \frac{1}{c} \cdot a}_{-2}+\underbrace{\rho \cdot \frac{1}{c} }_{1}\cdot y$

Thus $b-\rho \cdot \frac{1}{c} \cdot a=-2$ and $\rho \cdot \frac{1}{c}=1 \quad (2a)$

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To calculate the values of $\rho$ you use $1a$ and $2a$.