10.1.1 Conditional Distributions

The conditional distribution of $Y$ given $X=x$ can be found using the formula in Section 5.7:

Messy, but straight-forward, calculations will show that the conditional distribution is again normal with expectation

and variance ${\sigma }_{Y|x}^2={\sigma }_Y^2(1-{\rho }^2)$,

An easier and more elegant way of deriving this result is to use the transformation of $(U,V)$. Let us find the conditional distribution of $Y$ given $X=x$ again using this method. Since $X={\mu }_X+{\sigma }_{X}U$ (10.2) the condition $X=x$ corresponds to the condition $U=(x-{\mu }_X)/{\sigma }_X$ and inserting this into the expression for $Y$ (10.3) gives

which shows directly that

Notice that the conditional expectation is linear while the conditional variance is constant (does not depend on $x$).

Also note how the conditional variance depends on $\rho$. For $\rho$ close to $\pm 1$ the term $1-{\rho }^2$ is close to $0$ and the conditional variance thus small. This is saying that when $X$ and $Y$ are very correlated knowing $X$ gives us a lot of information about $Y$.

Representing $Y$ as a linear transformation of $X$ plus an independent normal random variable (${[(1-{\rho }^2){\sigma }_Y^2]}^{1/2}V$ above) is called regressing $Y$ on $X$ and is the idea behind the regression models you will meet in Math 235.

The conditional distribution of $X$ given $Y=y$ can be found analogously to be