Variance of sum of independent random variables

Dependencies:

  1. Variance of a random variable
  2. Independence of random variables (incomplete)
  3. Linearity of expectation
  4. Expectation of product of independent random variables (incomplete)

Let $X_1, X_2, \ldots, X_n$ be independent random variables. Then \[ \Var\left(\sum_{i=1}^n X_i\right) = \sum_{i=1}^n \Var(X_i) \]

Proof

This is trivially true when $n$ is 0 or 1.

For $n = 2$, we have \begin{align} \Var(X_1 + X_2) &= \E((X_1 + X_2)^2) - (\E(X_1 + X_2))^2 \\ &= (\E(X_1^2) + \E(X_2^2) + 2\E(X_1X_2)) \\ &\qquad - (\E(X_1)^2 + \E(X_2)^2 + 2\E(X_1)\E(X_2)) \tag{linearity of expectation} \\ &= \Var(X_1) + \Var(X_2) + 2(\E(X_1X_2) - \E(X_1)\E(X_2)) \\ &= \Var(X_1) + \Var(X_2) \tag{$X_1$ and $X_2$ are independent} \end{align}

For $n \ge 3$, we can prove by induction. \begin{align} & \Var\left(\sum_{i=1}^n X_i\right) \\ &= \Var\left(\sum_{i=1}^{n-1} X_i\right) + \Var(X_n) \tag{using induction hypothesis for $n=2$} \\ &= \left(\sum_{i=1}^{n-1} \Var(X_i)\right) + \Var(X_n) \tag{using induction hypothesis for $n-1$} \\ &= \sum_{i=1}^n \Var(X_i) \end{align}

Dependency for: None

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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. Conditional probability (incomplete)
  15. Independence of events
  16. Independence of composite events
  17. Generated σ-algebra
  18. Measurable function
  19. Borel algebra
  20. Generators of the real Borel algebra (incomplete)
  21. Random variable
  22. Expected value of a random variable
  23. Linearity of expectation
  24. Variance of a random variable
  25. Independence of random variables (incomplete)
  26. Expectation of product of independent random variables (incomplete)