Multivariate Normal Distribution Basics for Exam P

Bivariate Normal Distribution

The bivariate normal distribution for (X, Y) is characterized by five parameters: mu_X, mu_Y, sigma_X^2, sigma_Y^2, and the correlation rho. The joint PDF involves a complex formula, but for Exam P, the key properties matter more than the PDF itself.

Marginals: X ~ Normal(mu_X, sigma_X^2) and Y ~ Normal(mu_Y, sigma_Y^2). Any linear combination aX + bY is normal. Most importantly, for jointly normal random variables, uncorrelated (rho = 0) implies independent. This equivalence does not hold for general joint distributions.

Conditional Distributions

For bivariate normal (X, Y), the conditional distribution of Y given X = x is normal: Y | X = x ~ Normal(mu_Y + rho*(sigma_Y/sigma_X)*(x - mu_X), sigma_Y^2*(1 - rho^2)). The conditional mean is a linear function of x, and the conditional variance does not depend on x.

This result enables quick calculations. For example, if X and Y are bivariate normal with mu_X = 100, mu_Y = 80, sigma_X = 10, sigma_Y = 15, and rho = 0.6, then E[Y|X=120] = 80 + 0.6*(15/10)*(120 - 100) = 80 + 18 = 98, and Var(Y|X=120) = 225*(1 - 0.36) = 144.

Exam P Relevance

Exam P tests bivariate normal at a basic level. Key facts to remember: all marginals and conditionals are normal, linear combinations are normal, uncorrelated implies independent, and the conditional mean is the best linear predictor of Y given X. These properties are unique to the normal distribution and should be verified before applying them in problems that do not specify normality.