---
tag: 810WD1Q1WF
kind: proof
name: Proof of variance under translation
msc: [60A05]
proves: 04CYN5G6RY
uses: [007QED3E0Y, 0P7PTC8DDE]
sources:
  - url: https://en.wikipedia.org/wiki/Variance
    locator: Addition and multiplication by a constant
---

## Proof

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

1. Let $X$ be an integrable real random variable, let $b \in \R$, and let $m = E[X]$.
2. The constant $b$ is integrable with $E[b] = b$. By [[linearity of expectation|0P7PTC8DDE]], $X + b$ is integrable and $E[X + b] = m + b$.
3. Pointwise, $(X + b) - (m + b) = X - m$.
4. So $((X + b) - E[X + b])^2 = (X - m)^2$ as random variables.
5. By the definition of the [[variance|007QED3E0Y]], $\operatorname{Var}(X + b) = E[(X - m)^2] = \operatorname{Var}(X)$, as an equality in $[0, \infty]$.

## Idea

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