Not only are the marginal distributions of separate variables in the bivariate normal distribution also normally distributed, but the conditional distributions are normal too.
Conditional normal distributions
If \((X,Y) \sim \NormalDistn(\mu_X, \sigma_X^2, \mu_Y, \sigma_Y^2, \rho)\) then the conditional distribution of \(Y\), given \(X=x\) is univariate normal,
\[ \NormalDistn\left(\mu_Y + \frac{\sigma_Y}{\sigma_X}\rho(x-\mu_X),\; (1-\rho^2)\sigma_Y^2\right) \]We again first prove this result for the standard bivariate normal distribution (where \(\mu_X = \mu_Y = 0\) and \(\sigma_X^2 = \sigma_Y^2 = 1\)).
In the proof on the previous page, we showed that
\[ f(x,y) \;\;=\;\; \frac{1}{\sqrt{2\pi}} \exp\left(-\frac{x^2}2\right) \times \frac{1}{\sqrt{2\pi}\sqrt{1 - \rho^2}} \exp\left(-\frac{(y-\rho x)^2}{2(1-\rho^2)}\right) \]and that the marginal pdf of \(X\) is normal,
\[ f_X(x) \;\;=\;\; \frac{1}{\sqrt{2\pi}} \exp\left(-\frac{x^2}2\right) \]The conditional pdf of \(Y\) given \(X=x\) is therefore
\[ \begin{align} f_{Y \mid X=x}(y) \;\;&=\;\; \frac{f(x,y)}{f_X(x)} \\ &\;\; \frac{1}{\sqrt{2\pi}\sqrt{1 - \rho^2}} \exp\left(-\frac{(y-\rho x)^2}{2(1-\rho^2)}\right) \end{align} \]and this is the pdf of a \(\NormalDistn(\rho x, (1-\rho^2))\) distribution.
The general result can be obtained in a similar way using the joint and marginal pdfs of the general bivariate normal distribution.
Alternatively we could treat the general distribution as that of \(Y^* = \mu_Y + \sigma_Y Y\) and \(X^* = \mu_X + \sigma_X X\). The distribution of \(Y^* = \mu_Y + \sigma_Y Y\) would be
\[ \NormalDistn(\mu_Y + \sigma_Y\rho x, (1-\rho^2)\sigma_Y^2) \]Similarly, replacing the standardised value \(x\) with \(\frac{x^* - \mu_X}{\sigma_X}\) would give the general conditional distribution.
A similar result holds for the conditional distribution of \(X\), given that \(Y=y\).
Illustration
The following probability density function again describes two variables,
\[ (X,Y) \;\;\sim\;\; \NormalDistn(\mu_X=2, \sigma_X^2=1, \mu_Y=0, \sigma_Y^2=1, \rho) \]The slider under the diagram can adjust the value of the parameter \(\rho\).
Select Y from the pop-up menu to show a slice through the surface at \(y=-1.00\). The shape of this curve is the conditional distribution of \(X\), given that \(Y=-1.00\). The actual conditional pdf is this curve scaled to have unit area under it.
Drag the slider to observe the conditional distributions for other values of the variable \(Y\). When \(\rho = 0\), the conditional distribution of \(X\) does not depend on the value \(y\).
Now use the slider to increase \(\rho\) to 0.8 and again drag the righthand slider to see how the value of \(Y\) affects the conditional distribution of \(X\). Observe that