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)