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Lesson 3 of 12 · 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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Variance and covariance shortcuts, and the law of large numbers · Statistical Distributions for Financial Asset Prices and Returns