Variance of a random variable

Dependencies:

  1. Random variable
  2. Expected value of a random variable
  3. Linearity of expectation

Let $X$ be a real random variable. $\newcommand{\E}{\operatorname{E}}$ $\newcommand{\Var}{\operatorname{Var}}$ Then the variance of $X$, denoted as $\Var(X)$, is $\E((X-\E(X))^2)$.

An equivalent definition is $\Var(X) = \E(X^2) - \E(X)^2$.

Proof

\begin{align} \Var(X) &= \E((X-\E(X))^2) \\ &= \E(X^2 - 2\E(X)X + \E(X)^2) \\ &= \E(X^2) - 2\E(X)\E(X) + \E(X)^2 \tag{linearity of expectation} \\ &= \E(X^2) - \E(X)^2 \end{align}

Dependency for:

  1. Chebyshev's inequality
  2. Cantelli's inequality
  3. |mean - median| ≤ stddev
  4. Variance of sum of independent random variables
  5. Conditional variance
  6. Var(Y) = Var(E(Y|X)) + E(Var(Y|X))
  7. Var(aX + b) = a^2 Var(X)
  8. Normal distribution

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