10.1 The Bivariate Normal Distribution

Two continuous random variables $X$ and $Y$ are said to have a bivariate normal distribution with parameters $𝜽=({\mu }_X,{\mu }_Y,{\sigma }_X^2,{\sigma }_Y^2,\rho )$, where ${\sigma }_X^2>0$, ${\sigma }_Y^2>0$ and , if their joint pdf is given for all $x$ and $y$ by

$f_{XY}(x,y;𝜽)={\displaystyle \frac{1}{2\pi \sqrt{{\sigma }_X^2{\sigma }_Y^2(1-{\rho }^2)}}}\times \exp \left\{-{\displaystyle \frac{1}{2}}{\displaystyle \frac{1}{1-{\rho }^2}}Q(x,y)\right\},$

where

$Q(x,y)={\left({\displaystyle \frac{x-{\mu }_X}{{\sigma }_X}}\right)}^2-2\rho \left({\displaystyle \frac{x-{\mu }_X}{{\sigma }_X}}\right)\left({\displaystyle \frac{y-{\mu }_Y}{{\sigma }_Y}}\right)+{\left({\displaystyle \frac{y-{\mu }_Y}{{\sigma }_Y}}\right)}^2.$

It will be shown below that the marginal distributions of $X$ and $Y$ are both normal with $X\sim N({\mu }_X,{\sigma }_X^2)$, $Y\sim N({\mu }_Y,{\sigma }_Y^2)$ and $\mathrm{𝖢𝗈𝗋𝗋}[X,Y]=\rho$. This explains the notation and the restrictions on the parameters.

In the special case of $\rho =0$, i.e. no correlation, the joint pdf factorises into:

$f_{XY}(x,y;𝜽)={\displaystyle \frac{1}{\sqrt{2\pi {\sigma }_X^2}}}\exp \left\{-{\displaystyle \frac{1}{2}}{\left({\displaystyle \frac{x-{\mu }_X}{{\sigma }_X}}\right)}^2\right\}\times {\displaystyle \frac{1}{\sqrt{2\pi {\sigma }_Y^2}}}\exp \left\{-{\displaystyle \frac{1}{2}}{\left({\displaystyle \frac{y-{\mu }_Y}{{\sigma }_Y}}\right)}^2\right\},$

and so $(X,Y)$ are independent. That is, for bivariate normal random variables, no correlation implies independence. Recall from Section 7.1 that this is NOT true in general.

Realisations from this bivariate distribution and contour plots of the corresponding pdfs are shown on Figure 10.1 (First Link, Second Link), Figure 10.2 (First Link, Second Link) and Figure 10.3 (First Link, Second Link) for ${\mu }_X={\mu }_Y=0$ and ${\sigma }_X^2={\sigma }_Y^2=1$ and varying values of $\rho$. Note that the contours of the pdf are ellipses centred at the origin with orientation given by $\rho$. In general the contours will be ellipses centred at $({\mu }_X,{\mu }_Y)$.

The pdf is more conveniently expressed in matrix notation, which also makes the analogy with the univariate case clearer and the extension to higher dimensions easier. Set

so that

Then

Generating the bivariate normal To derive properties of the bivariate normal distribution it is often a good idea to think of it as a transformation of two independent standard normal random variables $U$ and $V$, say.

Now consider the linear transformation

$\left[\begin{array}{c}\hfill X\hfill \\ \hfill Y\hfill \end{array}\right]=\left[\begin{array}{c}\hfill {\mu }_X\hfill \\ \hfill {\mu }_Y\hfill \end{array}\right]+A\left[\begin{array}{c}\hfill U\hfill \\ \hfill V\hfill \end{array}\right],$

(10.1)

where

Or equivalently

$X={\mu }_X+{\sigma }_{X}U$

(10.2)

$Y={\mu }_Y+\rho {\sigma }_{Y}U+\left(\sqrt{1-{\rho }^2}\right){\sigma }_{Y}V.$

(10.3)

Clearly

1. $𝖤\left[X\right]={\mu }_X+{\sigma }_X𝖤\left[U\right]={\mu }_X$,

2. $𝖤\left[Y\right]={\mu }_Y+\rho {\sigma }_Y𝖤\left[U\right]+\left(\sqrt{1-{\rho }^2}\right){\sigma }_Y𝖤\left[V\right]={\mu }_Y$.

Using the independence of $U$ and $V$, the variances are

1. $\mathrm{𝖵𝖺𝗋}\left[X\right]={\sigma }_X^2\mathrm{𝖵𝖺𝗋}\left[U\right]={\sigma }_X^2$,

2. $\mathrm{𝖵𝖺𝗋}\left[Y\right]={\rho }^2{\sigma }_Y^2\mathrm{𝖵𝖺𝗋}\left[U\right]+\left(1-{\rho }^2\right){\sigma }_Y^2\mathrm{𝖵𝖺𝗋}\left[V\right]={\sigma }_Y^2$.

Finally, the covariance and correlation are (again using independence of $U$ and $V$)

	$\mathrm{𝖢𝗈𝗏}[X,Y]$	$=\mathrm{𝖢𝗈𝗏}[{\sigma }_{X}U,\rho {\sigma }_{Y}U]+\mathrm{𝖢𝗈𝗏}[{\sigma }_{X}U,\sqrt{1-{\rho }^2}V]$
		$=\rho {\sigma }_X{\sigma }_Y\mathrm{𝖵𝖺𝗋}\left[U\right]=\rho {\sigma }_X{\sigma }_Y$

and

$\mathrm{𝖢𝗈𝗋𝗋}[X,Y]=\rho .$

We have shown that the parameters have the intuitive interpretation. We also need to show that $(X,Y)$ have the correct joint distribution. We do this for a general 1-1 linear transformation of a vector of $d$ iid $𝖭(0,1)$ rvs (i.e. not just for $2$).