Lesson 3 of 14 · All lessons in this module
  1. Random variables: PMF, PDF and CDF
  2. Expected value, variance and standard deviationVideo lesson
  3. Variance and covariance shortcuts, and the law of large numbers
  4. Discrete distributions: uniform, binomial and Poisson
  5. Continuous uniform and normal distributions, and simulating random draws
  6. Log-normal and logistic distributions, and the moments of key distributions
  7. Integrated cases: reading probabilities and moments from a return model
  8. Conditional probability and probability trees
  9. Conditional expected values and variancesVideo lesson
  10. Covariance from a joint probability function
  11. Joint, marginal and conditional distributions as information arrives
  12. Bayes' formula: updating a probability with new informationVideo lesson
  13. Bayes in practice: diffuse priors, missing inputs and screening tests
  14. Integrated cases: a tree with model forecasts and a Bayes update

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Lesson 3 of 14 · 14 min

Variance and covariance shortcuts, and the law of large numbers

Variance is the mean of the square minus the square of the mean, covariance is the mean of the product minus the product of the means, and with enough random observations a sample mean settles on the true mean.

In short

  • Unconditional mean, variance and covariance describe variables before you condition on any scenario. They are the moments of the distribution.
  • Variance shortcut: Var(X)=E[X2]−(E[X])2\text{Var}(X) = E[X^2] - (E[X])^2.
  • Covariance shortcut: Cov(X,Y)=E[XY]−E[X]E[Y]\text{Cov}(X,Y) = E[XY] - E[X]E[Y]. For independent variables E[XY]=E[X]E[Y]E[XY] = E[X]E[Y], so the covariance is zero.
  • The sign of a covariance gives the direction of a linear relationship; its size depends on units, so a large covariance alone does not prove a strong relationship.
  • (x−μ)/σ(x - \mu)/\sigma tells you how many standard deviations an outcome lies from the mean.
  • Law of large numbers: as a random, representative sample grows, its mean converges to the true (unconditional) mean.

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