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$E[g(X)] = \int g(X)\ dP \nonumber$

Suponemos, sin aseveración repetida, que las variables aleatorias y las funciones Borel de variables aleatorias o vectores aleatorios son integrables. El uso de una expresión como$$I_M (X)$$ implica la suposición tácita de que$$M$$ es un conjunto de Borel en el codominio de$$X$$.
(E1):$$E[aI_A] = aP(A)$$, cualquier constante$$a$$, cualquier evento$$A$$
(E1a):$$E[I_M (X)] = P(X \in M)$$ y$$E[I_M (X) I_N (Y)] - P(X \in M, Y \in N)$$ para cualquier conjunto de Borel$$M, N$$ (Se extiende a cualquier producto finito de tales funciones indicadoras de vectores aleatorios)
(E2): Linealidad. Para cualquier constante$$a, b$$,$$E[aX + bY) = aE[X] + bE[Y]$$ (Se extiende a cualquier combinación lineal finita)
a.$$X \ge 0$$ a.s. implica$$E[X] \ge 0$$, con igualdad iff$$X = 0$$ a.s.
b.$$X \ge Y$$ a.s. implica$$E[X] \ge E[Y]$$, con igualdad iff$$X = Y$$ a.s.
(E4): Lema fundamental. Si$$X \ge 0$$ está acotado, y$$\{X_n: 1 \le n\}$$ es a.s. no negativo, no decreciente, con$$\text{lim}_n X_n (\omega) \ge X(\omega)$$ para a.e.$$\omega$$, entonces$$\text{lim}_n E[X_n] \ge E[X]$$
(E4a): Convergencia monótona. Si para todos$$n$$,$$0 \le X_n \le X_{n + 1}$$ a.s. y$$X_n \to X$$ a.s., entonces$$E[X_n] \to E[X]$$ (El teorema también sostiene si$$E[X] = \infty$$)

******
(E5): Unicidad. * debe leerse como uno de los símbolos$$\le, =$$, o$$\ge$$
a.$$E[I_M(X) g(X)]$$ *$$E[I_M(X) h(X)]$$ para todos los$$M$$ iff$$g(X)$$ *$$h(X)$$ a.s.
b.$$E[I_M(X) I_N (Z) g(X, Z)] = E[I_M (X) I_N (Z) h(X,Z)]$$ para todos los$$M, N$$ iff$$g(X, Z) = h(X, Z)$$ a.s.
(E6): Lema de Fatou. Si$$X_n \ge 0$$ a.s., para todos$$n$$, entonces$$E[ \text{lim inf } X_n] \le [\text{lim inf } E[X_n]$$
(E7): Convergencia dominada. Si$$X_n \to X$$ a.s reales o complejos,$$|X_n| \le Y$$ a.s. para todos$$n$$, y$$Y$$ es integrable, entonces$$\text{lim}_n E[X_n] = E[X]$$
a. Si$$X$$ es integrable sobre$$E$$, y$$E = \bigvee_{i = 1}^{\infty} E_i$$ (unión disjunta), entonces$$E[I_E X] = \sum_{i = 1}^{\infty} E[I_{E_i} X]$$
b. Si$$\sum_{n = 1}^{\infty} E[|X_n|] < \infty$$, entonces$$\sum_{n = 1}^{\infty} |X_n| < \infty$$, a.s. y$$E[\sum_{n = 1}^{\infty} X_n] = \sum_{n = 1}^{\infty} E[X_n]$$
a. $$X$$es integrable iff ambos$$X^{+}$$ y$$X^{-}$$ son integrables iff$$|X|$$ es integrable.
b.$$X$$ es integrable iff$$E[I_{\{|X| > a\}} |X|] \to 0$$ como$$a \to \infty$$
c. Si$$X$$ es integrable, entonces$$X$$ es a.s. finito
d. Si$$E[X]$$ existe y$$P(A) = 0$$, entonces$$E[I_A X] = 0$$
(E10): Triángulo desigualdad. Para integrable$$X$$, real o complejo,$$|E[X]| \le E[|X|]$$
(E11): Teorema del valor medio. Si$$a \le X \le b$$ a.s. on$$A$$, entonces$$aP(A) \le E[I_A X] \le bP(A)$$
(E12): Para no negativos, Borel$$g$$,$$E[g(X)] \ge aP(g(X) \ge a)$$
(E13): La desigualdad de Markov. Si$$g \ge 0$$ y no decreciente para$$t \ge 0$$ y$$a \ge 0$$, entonces

$$g(a)P(|X| \ge a) \le E[g(|X|)]$$

(E14): La desigualdad de Jensen. Si$$g$$ es convexo en un intervalo que contiene el rango de variable aleatoria$$X$$, entonces$$g(E[X]) \le E[g(X)]$$
(E15): Desigualdad de Schwarz. Para$$X, Y$$ real o complejo,$$|E[XY]|^2 \le E[|X|^2] E[|Y|^2]$$, con igualdad iff hay una constante$$c$$ tal que$$X = cY$$ a.s.
(E16): la desigualdad de Hölder. Para$$1 \le p, q$$, con$$\dfrac{1}{p} + \dfrac{1}{q} = 1$$, y$$X, Y$$ real o complejo.

$$E[|XY|] \le E[|X|^p]^{1/p} E[|Y|^q]^{1/q}$$

(E17): La desigualdad de Hölder. Para$$1 < p$$ y$$X, Y$$ real o complejo,

$$E[|X + Y|^p]^{1/p} \le E[|X|^p]^{1/p} + E[|Y|^p]^{1/p}$$

(E18): Independencia y expectativa. Las siguientes condiciones son equivalentes.
a. el par$$\{X, Y\}$$ es independiente
b.$$E[I_M (X) I_N (Y)] = E[I_M (X)] E[I_N (Y)]$$ para todos Borel$$M, N$$
c.$$E[g(X)h(Y)] = E[g(X)] E[h(Y)]$$ para todos Borel$$g, h$$ tal que$$g(X)$$,$$h(Y)$$ son integrables.
(E19): Caso especial del teorema de Radón-Nikodym Si$$g(Y)$$ es integrable y$$X$$ es un vector aleatorio, entonces existe una función Borel de valor real$$e(\cdot)$$, definida en el rango de$$X$$, a.s únicas$$[P_X]$$, tal que$$E[I_M(X) g(X)] = E[I_M (X) e(X)]$$ para todos Borel establece$$M$$ en el codominio de$$X$$.
(E20): Algunas formas especiales de expectativa
a. Supongamos que$$F$$ es no decreciente, derecha continua en$$[0, \infty)$$, con$$F(0^{-}) = 0$$. Vamos$$F^{*} (t) = F(t - 0)$$. Considera$$X \ge 0$$ con$$E[F(X)] < \infty$$. Entonces,

(1)$$E[F(X)] = \int_{0}^{\infty} P(X \ge t) F\ (dt)$$ y (2)$$E[F^{*} (X)] = \int_{0}^{\infty} P(X > t) F\ (dt)$$

b. Si$$X$$ es integrable, entonces$$E[X] = \int_{-\infty}^{\infty} [u(t) - F_X (t)]\ dt$$
c. Si$$X, Y$$ son integrables, entonces$$E[X - Y] = \int_{-\infty}^{\infty} [F_Y (t) - F_X (t)]\ dt$$
d. si$$X \ge 0$$ es integrable, entonces

$$\sum_{n = 0}^{\infty} P(X \ge n + 1) \le E[X] \le \sum_{n = 0}^{\infty} P(X \ge n) \le N \sum_{k = 0}^{\infty} P(X \ge kN)$$, para todos$$N \ge 1$$

e. Si integrable$$X \ge 0$$ es de valor entero, entonces

$$E[X] = \sum_{n = 1}^{\infty} P(X \ge n) = \sum_{n = 0}^{\infty} P(X > n) E[X^2] = \sum_{n = 1}^{\infty} (2n - 1) P(X \ge n) = \sum_{n = 0}^{\infty} (2n + 1) P(X > n)$$

f. Si$$Q$$ es la función quantile for$$F_X$$, entonces$$E[g(X)] = \int_{0}^{1} g[Q(u)]\ du$$

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