Math 2.0

Probability theory and stochastic processes Proof

Proof of variance under translation

Builds on VarianceLinearity of expectation

Proof

We show that the deviation from the mean does not change.

  1. Let XX be an integrable real random variable, let b∈ℝb \in \R, and let m=E[X]m = E[X].
  2. The constant bb is integrable with E[b]=bE[b] = b. By linearity of expectation, X+bX + b is integrable and E[X+b]=m+bE[X + b] = m + b.
  3. Pointwise, (X+b)−(m+b)=X−m(X + b) - (m + b) = X - m.
  4. So ((X+b)−E[X+b])2=(X−m)2((X + b) - E[X + b])^2 = (X - m)^2 as random variables.
  5. By the definition of the variance, Var⁡(X+b)=E[(X−m)2]=Var⁡(X)\operatorname{Var}(X + b) = E[(X - m)^2] = \operatorname{Var}(X), as an equality in [0,∞][0, \infty].

Idea

Adding bb moves every value and the mean by the same amount. The distances to the mean stay the same, so the variance stays the same.

Reading path

To understand this page from the ground up, read these 46 pages in order. The foundations come first.

  1. Construction of the reals from Cauchy sequences concept
  2. Element definition
  3. Integers object
  4. Ordered field definition
  5. Proposition concept
  6. Random experiment concept
  7. Rational numbers object
  8. Real numbers object
  9. Set definition
  10. Ring of integers statement
  11. Disjunction definition
  12. Implication definition
  13. Predicate definition
  14. Ordered pair definition
  15. Union definition
  16. Sample space definition
  17. The rationals form a field statement
  18. Existential quantifier definition
  19. Universal quantifier definition
  20. Cartesian product definition
  21. Subset definition
  22. The rationals form an ordered field statement
  23. Uniqueness quantifier definition
  24. Binary relation definition
  25. Universal set definition
  26. Order on the reals definition
  27. Power set definition
  28. Complement definition
  29. Function definition
  30. Interval definition
  31. Injective function definition
  32. Preimage of a set definition
  33. Sigma-algebra definition
  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. Proof of variance under translation proof

Where this leads

2 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Variance, Addition and multiplication by a constant
About this page
Tag810WD1Q1WF
URIm2://810WD1Q1WF
Kindproof
WritingW0 stub
MSC60A05
ProvesVariance under translation
UsesVariance, Linearity of expectation
Changed4 November 2026

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