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:

    • 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).