Var(aX + b) = a^2 Var(X)

Dependencies:

  1. Variance of a random variable
  2. Linearity of expectation

Let $X$ be a random variable. Let $a \in \mathbb{R}$ and $b \in \operatorname{support}(X)$. Then $\newcommand{\E}{\operatorname{E}}\newcommand{\Var}{\operatorname{Var}}$ $\Var(aX + b) = a^2\Var(X)$.

Proof

\[ \Var(aX + b) = \E((aX + b - \E(aX + b))^2) = \E((aX - a\E(X))^2) = a^2\E((X - \E(X))^2) = a^2\Var(X) \]

Dependency for: None

Info:

Transitive dependencies:

  1. /measure-theory/linearity-of-lebesgue-integral
  2. /measure-theory/lebesgue-integral
  3. /sets-and-relations/de-morgan-laws
  4. /sets-and-relations/countable-set
  5. /analysis/topological-space
  6. Group
  7. Ring
  8. Field
  9. Vector Space
  10. σ-algebra
  11. σ-algebra is closed under countable intersections
  12. Measure
  13. Probability
  14. Generated σ-algebra
  15. Measurable function
  16. Borel algebra
  17. Generators of the real Borel algebra (incomplete)
  18. Random variable
  19. Expected value of a random variable
  20. Linearity of expectation
  21. Variance of a random variable