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:
- If
x ~ N(0, σx²)
andy ~ N(0, σy²)
, thenx + y ~ N(0, σx² + σy²)
- If
Central Limit Theorem:
- The 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 the 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 the possibility of asymmetric distributions.
- Fat-tailed distributions imply larger probability of worse-case scenarios (risk management).