17.5: Appendix E to Applied Probability - Properties of Mathematical Expectation
\[E[g(X)] = \int g(X)\ dP \nonumber\]
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 \(E[g(X)] = \int_{0}^{1} g[Q(u)]\ du\)