Linearity of expectation

Dependencies:

  1. Vector Space
  2. Probability
  3. Random variable
  4. Expected value of a random variable
  5. /measure-theory/linearity-of-lebesgue-integral

Let $V$ be a vector space over $\mathbb{R}$. Let $X_1, X_2, \ldots, X_n$ be random variables in $V$ over the probability space $(\Omega, \mathcal{F}, \Pr)$. Let $b, a_1, a_2, \ldots, a_n$ be real numbers. $\newcommand{\E}{\operatorname{E}}$

Let $Y = b + \sum_{i=1}^n a_iX_i$. Then \[ \E(Y) = b + \sum_{i=1}^n a_i\E(X_i) \]

Proof

\begin{align} \E(Y) &= \int_{\omega \subseteq \Omega} Y(\omega) \Pr(\omega) \\ &= \int_{\omega \subseteq \Omega} \left(b + \sum_{i=1}^n a_iX_i(\omega)\right) \Pr(\omega) \\ &= b \left( \int_{\omega \subseteq \Omega} \Pr(\omega) \right) + \sum_{i=1}^n a_i \left(\int_{\omega \subseteq \Omega}X_i(\omega) \Pr(\omega)\right) \tag{linearity of Lebesgue integral} \\ &= b + \sum_{i=1}^n a_i \E(X_i) \end{align}

Dependency for:

  1. Cantelli's inequality
  2. Chernoff bound
  3. Cauchy-Schwarz inequality for random variables
  4. Linearity of expectation for matrices
  5. Variance of sum of independent random variables
  6. Var(aX + b) = a^2 Var(X)
  7. Var(Y) = Var(E(Y|X)) + E(Var(Y|X))
  8. Covariance of 2 random variables
  9. Variance of a random variable
  10. |mean - median| ≤ stddev
  11. Minimizer of f(z) = E(|X-z|) is median
  12. Normal distribution
  13. Markov chains: recurrent iff expected number of visits is infinite

Info:

Transitive dependencies:

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