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 $X_1,X_2,\dots ,X_n$ be random variables in $V$ over the probability space $(\mathrm{\Omega },\mathcal{F},Pr)$. Let $b,a_1,a_2,\dots ,a_n$ be real numbers. $$

Let $Y=b+\sum _{i=1}^n{a}_i{X}_i$. Then $$\mathrm{E}(Y)=b+\sum _{i=1}^n{a}_i\mathrm{E}(X_i)$$

Proof

$$\begin{array}{rl}\mathrm{E}(Y)& ={\int }_{\omega \subseteq \mathrm{\Omega }}Y(\omega )Pr(\omega )\\ & ={\int }_{\omega \subseteq \mathrm{\Omega }}(b+\sum _{i=1}^n{a}_i{X}_i(\omega ))Pr(\omega )\\ \text{(linearity of Lebesgue integral)}& & =b({\int }_{\omega \subseteq \mathrm{\Omega }}Pr(\omega ))+\sum _{i=1}^n{a}_i({\int }_{\omega \subseteq \mathrm{\Omega }}{X}_i(\omega )Pr(\omega ))& =b+\sum _{i=1}^n{a}_i\mathrm{E}(X_i)\end{array}$$

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:

• Depth: 7
• Number of transitive dependencies: 19

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