# Probability, statistical distributions, descriptive statistics

Normal distributions

• Normally distributed random variable with mean μ and variance σ2:

• All higher order moments are given in terms of μ and variance σ2:

• Easily manipulated:

– ifx~N(0,σx2)andy~N(0,σy2),thenx+y~N(0,σx2+σy2)

• Central limit theorem: sum of a large number of IID random variables (with

finite mean and variance) is normally distributed

• For linear models:

– Normal distributions are preserved by principle of superposition

– Normally distributed forecast errors: Maximum likelihood gives least squares

– Useful for calculating prediction intervals

• Problems for nonlinear systems:

– Use of normal distributions neglects possibility of asymmetric distributions

– Fat tailed distributions imply larger probability of worse case scenarios (risk

management)