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) and y ~ N(0, σy²), then x + y ~ N(0, σx² + σy²)

• 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).