Expected value of a random variable

Dependencies:

1. Probability
2. Random variable
3. Vector Space
4. /measure-theory/lebesgue-integral

Let $(\Omega, \mathcal{F}, \Pr)$ be a probability space. Let $V$ be a vector space over $\mathbb{R}$ and let $D \subseteq V$. Let $X: \Omega \mapsto D$ be a random variable.

Then the expected value of $X$, denoted as $\operatorname{E}(X)$, is \[ \operatorname{E}(X) = \int_{\omega \subseteq \Omega} X(\omega) \Pr(\omega) \] The integral above is the Lebesgue integral.

There are easier-to-comprehend equivalent definitions for simpler cases:

• When $D$ is countable, \[ \operatorname{E}(X) = \sum_{x \in D} x \Pr(X = x) \]
• When $D = \mathbb{R}$, \[ \operatorname{E}(X) = \int_{-\infty}^{\infty} x f_X(x) \mathrm{d}x \] Here $f_X$ is the probability density function of $X$.

It is easy to see that if $X$ is a matrix, then $\operatorname{E}(X)_{i, j} = \operatorname{E}(X_{i, j})$.

Dependency for:

1. Chebyshev's inequality
2. Markov's bound
3. Cantelli's inequality
4. Chernoff bound
5. Linearity of expectation
6. X ≤ Y ⟹ E(X) ≤ E(Y)
7. Cauchy-Schwarz inequality for random variables
8. Law of total probability: P(A) = E(P(A|X)) (incomplete)
9. Conditional expectation
10. Law of total probability: E(Y) = E(E(Y|X)) (incomplete)
11. Expectation of product of independent random variables (incomplete)
12. Var(Y) = Var(E(Y|X)) + E(Var(Y|X))
13. Covariance of 2 random variables
14. Conditional variance
15. Variance of a random variable
16. |mean - median| ≤ stddev
17. Minimizer of f(z) = E(|X-z|) is median
18. Normal distribution

Info:

• Depth: 6
• Number of transitive dependencies: 17

Transitive dependencies:

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