# 17.5: Appendix E to Applied Probability - Properties of Mathematical Expectation

• • Paul Pfeiffer
• Rice University
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$E[g(X)] = \int g(X)\ dP \nonumber$

We suppose, without repeated assertion, that the random variables and Borel functions of random variables or random vectors are integrable. Use of an expression such as $$I_M (X)$$ involves the tacit assumption that $$M$$ is a Borel set on the codomain of $$X$$.
(E1): $$E[aI_A] = aP(A)$$, any constant $$a$$, any event $$A$$
(E1a): $$E[I_M (X)] = P(X \in M)$$ and $$E[I_M (X) I_N (Y)] - P(X \in M, Y \in N)$$ for any Borel sets $$M, N$$ (Extends to any finite product of such indicator functions of random vectors)
(E2): Linearity. For any constants $$a, b$$, $$E[aX + bY) = aE[X] + bE[Y]$$ (Extends to any finite linear combination)
(E3): Positivity; monotonicity.
a. $$X \ge 0$$ a.s. implies $$E[X] \ge 0$$, with equality iff $$X = 0$$ a.s.
b. $$X \ge Y$$ a.s. implies $$E[X] \ge E[Y]$$, with equality iff $$X = Y$$ a.s.
(E4): Fundamental lemma. If $$X \ge 0$$ is bounded, and $$\{X_n: 1 \le n\}$$ is a.s. nonnegative, nondecreasing, with $$\text{lim}_n X_n (\omega) \ge X(\omega)$$ for a.e. $$\omega$$, then $$\text{lim}_n E[X_n] \ge E[X]$$
(E4a): Monotone convergence. If for all $$n$$, $$0 \le X_n \le X_{n + 1}$$ a.s. and $$X_n \to X$$ a.s.,then $$E[X_n] \to E[X]$$ (The theorem also holds if $$E[X] = \infty$$)

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(E5): Uniqueness. * is to be read as one of the symbols $$\le, =$$, or $$\ge$$
a. $$E[I_M(X) g(X)]$$ * $$E[I_M(X) h(X)]$$ for all $$M$$ iff $$g(X)$$ * $$h(X)$$ a.s.
b. $$E[I_M(X) I_N (Z) g(X, Z)] = E[I_M (X) I_N (Z) h(X,Z)]$$ for all $$M, N$$ iff $$g(X, Z) = h(X, Z)$$ a.s.
(E6): Fatou's lemma. If $$X_n \ge 0$$ a.s., for all $$n$$, then $$E[ \text{lim inf } X_n] \le [\text{lim inf } E[X_n]$$
(E7): Dominated convergence. If real or complex $$X_n \to X$$ a.s., $$|X_n| \le Y$$ a.s. for all $$n$$, and $$Y$$ is integrable, then $$\text{lim}_n E[X_n] = E[X]$$
(E8): Countable additivity and countable sums.
a. If $$X$$ is integrable over $$E$$, and $$E = \bigvee_{i = 1}^{\infty} E_i$$ (disjoint union), then $$E[I_E X] = \sum_{i = 1}^{\infty} E[I_{E_i} X]$$
b. If $$\sum_{n = 1}^{\infty} E[|X_n|] < \infty$$, then $$\sum_{n = 1}^{\infty} |X_n| < \infty$$, a.s. and $$E[\sum_{n = 1}^{\infty} X_n] = \sum_{n = 1}^{\infty} E[X_n]$$
(E9): Some integrability conditions
a. $$X$$ is integrable iff both $$X^{+}$$ and $$X^{-}$$ are integrable iff $$|X|$$ is integrable.
b. $$X$$ is integrable iff $$E[I_{\{|X| > a\}} |X|] \to 0$$ as $$a \to \infty$$
c. If $$X$$ is integrable, then $$X$$ is a.s. finite
d. If $$E[X]$$ exists and $$P(A) = 0$$, then $$E[I_A X] = 0$$
(E10): Triangle inequality. For integrable $$X$$, real or complex, $$|E[X]| \le E[|X|]$$
(E11): Mean-value theorem. If $$a \le X \le b$$ a.s. on $$A$$, then $$aP(A) \le E[I_A X] \le bP(A)$$
(E12): For nonnegative, Borel $$g$$, $$E[g(X)] \ge aP(g(X) \ge a)$$
(E13): Markov's inequality. If $$g \ge 0$$ and nondecreasing for $$t \ge 0$$ and $$a \ge 0$$, then

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

(E14): Jensen's inequality. If $$g$$ is convex on an interval which contains the range of random variable $$X$$, then $$g(E[X]) \le E[g(X)]$$
(E15): Schwarz' inequality. For $$X, Y$$ real or complex, $$|E[XY]|^2 \le E[|X|^2] E[|Y|^2]$$, with equality iff there is a constant $$c$$ such that $$X = cY$$ a.s.
(E16): Hölder's inequality. For $$1 \le p, q$$, with $$\dfrac{1}{p} + \dfrac{1}{q} = 1$$, and $$X, Y$$ real or complex.

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

(E17): Hölder's inequality. For $$1 < p$$ and $$X, Y$$ real or complex,

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

(E18): Independence and expectation. The following conditions are equivalent.
a. The pair $$\{X, Y\}$$ is independent
b. $$E[I_M (X) I_N (Y)] = E[I_M (X)] E[I_N (Y)]$$ for all Borel $$M, N$$
c. $$E[g(X)h(Y)] = E[g(X)] E[h(Y)]$$ for all Borel $$g, h$$ such that $$g(X)$$, $$h(Y)$$ are integrable.
(E19): Special case of the Radon-Nikodym theorem If $$g(Y)$$ is integrable and $$X$$ is a random vector, then there exists a real-valued Borel function $$e(\cdot)$$, defined on the range of $$X$$, unique a.s. $$[P_X]$$, such that $$E[I_M(X) g(X)] = E[I_M (X) e(X)]$$ for all Borel sets $$M$$ on the codomain of $$X$$.
(E20): Some special forms of expectation
a. Suppose $$F$$ is nondecreasing, right-continuous on $$[0, \infty)$$, with $$F(0^{-}) = 0$$. Let $$F^{*} (t) = F(t - 0)$$. Consider $$X \ge 0$$ with $$E[F(X)] < \infty$$. Then,

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

b. If $$X$$ is integrable, then $$E[X] = \int_{-\infty}^{\infty} [u(t) - F_X (t)]\ dt$$
c. If $$X, Y$$ are integrable, then $$E[X - Y] = \int_{-\infty}^{\infty} [F_Y (t) - F_X (t)]\ dt$$
d. if $$X \ge 0$$ is integrable, then

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

e. If integrable $$X \ge 0$$ is integer-valued, then

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

f. If $$Q$$ is the quantile function for $$F_X$$, then $$E[g(X)] = \int_{0}^{1} g[Q(u)]\ du$$

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