3.10  Normal and Related Distributions

In this section we describe families of distributions related to the normal distribution. Specific families and shorthand notation for each are

1. normal: N(μ,σ²),
2. lognormal: λ(μ,σ²),
3. chi-squared: χ²(ν,δ²),
4. joint-normal: N_n(μ,Σ).

Normal distributions are enormously important in probability and statistics. This is due to the central limit theorem, discussed in Section 3.15.

Lognormal distributions comprise a minor family of distributions that happen to arise frequently in finance. This is due to the exponential effect of compounding. If an asset’s value tends to be normally distributed, returns on that asset tend to be lognormally distributed.

Chi-squared distributions arrise when normal random variables are squared. They will be important in Section 10.3, where we consider quadratic transformations. These entail quadratic polynomials of normal random variables.

Joint-normal distributions (also called multivariate normal or multinormal distributions) extend normal distributions to multiple dimensons.