Math 2.0

Probability theory and stochastic processes Proposition

Variance as the mean of the square minus the square of the mean

Builds on VarianceLinearity of expectation

Statement

Let XX be a real random variable with E[X2]<∞E[X^2] < \infty. Then Var⁡(X)=E[X2]−(E[X])2.\operatorname{Var}(X) = E[X^2] - (E[X])^2.

Remarks

Proof: with μ=E[X]\mu = E[X], (X−μ)2=X2−2μX+μ2(X - \mu)^2 = X^2 - 2\mu X + \mu^2. By linearity of expectation, E[(X−μ)2]=E[X2]−2μ2+μ2=E[X2]−μ2E[(X - \mu)^2] = E[X^2] - 2\mu^2 + \mu^2 = E[X^2] - \mu^2. As a corollary, (E[X])2≤E[X2](E[X])^2 \le E[X^2]. For a fair die, E[X2]=91/6E[X^2] = 91/6 and E[X]=7/2E[X] = 7/2, so Var⁡(X)=91/6−49/4=35/12\operatorname{Var}(X) = 91/6 - 49/4 = 35/12. In floating-point arithmetic, this formula can lose much precision when E[X2]E[X^2] and (E[X])2(E[X])^2 are close.

Reading path

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  1. Construction of the reals from Cauchy sequences concept
  2. Element definition
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  7. Rational numbers object
  8. Real numbers object
  9. Set definition
  10. Ring of integers statement
  11. Disjunction definition
  12. Implication definition
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  15. Union definition
  16. Sample space definition
  17. The rationals form a field statement
  18. Existential quantifier definition
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  20. Cartesian product definition
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  22. The rationals form an ordered field statement
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  34. Countable set definition
  35. Event definition
  36. Borel sigma-algebra definition
  37. Probability axioms concept
  38. Probability measure definition
  39. Probability space definition
  40. Random variable definition
  41. Discrete random variable definition
  42. Random variable with a density definition
  43. Expected value definition
  44. Linearity of expectation statement
  45. Variance definition
  46. Variance as the mean of the square minus the square of the mean statement

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Further reading

  1. Web https://en.wikipedia.org/wiki/Variance, Definition
About this page
Tag0NMF95YK8W
URIm2://0NMF95YK8W
Aliasescomputational-formula-for-variance
Kindstatement, proposition
WritingW0 stub
MSC60A05
StatusThis copy of the site has no build data. Run m2 status 0NMF95YK8W.
UsesVariance, Linearity of expectation

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