A portfolio of jointly normal asset returns is the single most common modelling assumption in classical portfolio theory — and it is only tractable because the multivariate normal distribution, the joint generalisation of The Normal Distribution, has three remarkable closure properties: every marginal is normal, every linear combination is normal, and every conditional distribution is normal. This capstone lesson builds the bivariate normal density explicitly, states the general multivariate case, and derives the conditional-normality result that is the direct mathematical ancestor of linear regression and the Kalman filter.
Learning objectives. By the end of this note you should be able to:
- write the bivariate normal density in terms of the two means, two variances, and the correlation;
- state the closure properties of the multivariate normal family (marginals, linear combinations, conditionals all normal);
- compute the conditional mean and variance of one jointly normal variable given another;
- connect the correlation parameter to the shape of the joint density's contours.
In this lesson
- The bivariate normal density
- The general multivariate case and closure properties
- The conditional distribution — bivariate case
Before you start
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