17.6: Appendix F to Applied Probability- Properties of conditional expectation, given a random vector


We suppose, without repeated assertion, that the random variables and functions of random vectors are integrable, as needed.
(CE1): Defining condition. $$e(X) = E[g(Y)|X]$$ a.s. iff $$E[I_M (X) g(Y)] = E[I_M (X) e(X)]$$ for each Borel set $$M$$ on the codomain of $$X$$.
(CE1a): If $$P(X \in M) > 0$$, then $$E[I_M(X) e(X)] = E[g(Y)|X \in M] P(X \in M)$$
(CE1b): Law of total probability. $$E[g(Y)] = E\{[g(Y)|X]\}$$
(CE2): Linearity. For any constants $$a, b$$
$$E[ag(Y) + bh(Z)|X] = aE[g(Y)|X] + bE[h(Z)|X]$$ a.s.
(Extends to any finite linear combination)
(CE3): positivity; monotonicty.
a. $$g(Y) \ge 0$$ a.s. implies $$E[g(Y)|X] \ge 0$$ a.s.
b. $$g(Y) \ge h(Z)$$ a.s. implies $$E[g(Y)|X] \ge E[h(Z)|X]$$ a.s.
(CE4): Monotone convergence. $$Y_n \to Y$$ a.s. monotonically implies $$E[Y_n |X] \to E[Y|X]$$ a.s.
(CE5): Independence. $$\{X, Y\}$$ is an independent pair
a. iff $$E[g(Y)|X] = E[g(Y)]$$ a.s. for all Borel functions $$g$$
b. iff $$E[I_N (Y)|X] = E[I_N (Y)]$$ a.s. for all Borel sets $$N$$ on the codomain of $$Y$$
(CE6): $$e(X) = E[g(Y)|X]$$ a.s. iff $$E[h(X)g(Y)] = E[h(X)e(X)]$$ a.s. for any Borel function $$h$$
(CE7): $$E[h(X)|X] = h(X)$$ a.s. for any Borel function $$h$$
(CE8): $$E[h(X)g(Y)|X] = h(X) E[g(Y)|X]$$ a.s. for any Borel function $$h$$
(CE9): If $$X = h(W)$$ and $$W = k(X)$$, with $$h, k$$ Borel functions, then $$E[g(Y)|X] = E[g(Y)|W]$$ a.s.
(CE10): If $$g$$is a Borel function such that $$E[g(t, Y)]$$ is finite for all $$t$$ on the range of $$X$$ and $$E[g(X, Y)]$$ is finite, then
a. $$E[g(X, Y)|X = t] = E[g(t, Y)|X = t]$$ a.s. $$[P_X]$$
b. If $$\{X, Y\}$$ is independent, then $$E[g(X, Y)|X = t] = E[g(t, Y)]$$ a.s. $$[P_X]$$
(CE11): Suppose $$\{X(t): t \in T\}$$ is a real-valued measurable random process whose parameter set $$T$$ is a Borel subset of the real line and $$S$$ is a random variable whose range is a subset of $$T$$, so that $$X(S)$$ is a random variable.
If $$E[X(t)]$$ is finite for all $$t$$ in $$T$$ and $$E[X(S)]$$ is finite, then
a. \9E[X(S)|S = t] = E[X(t)|S = t]\) a.s $$[P_S]$$
b. If, in addition, $$\{S, X_T\}$$ is independent, then $$E[X(S)|S = t] = E[X(t)]$$ a.s. $$[P_S]$$
(CE12): Countable additivity and countable sums.
a. If $$Y$$ is integrable on $$A$$ and $$A = \bigvee_{n = 1}^{\infty} A_n$$.
then $$E[I_A Y|X] = \sum_{n = 1}^{\infty} E[I_A Y|X]$$ a.s.
b. If $$\sum_{n = 1}^{\infty} E[|Y_n|] < \infty$$, thne $$E[\sum_{n = 1}^{\infty} Y_n|X]$$ a.s.
(CE13): Triangle inequality. $$|E[g(Y)|X]| \le E[|g(Y)||X]$$ a.s.
(CE14): Jensen's inequality. If $$g$$ is a convex function on an interval $$I$$ which contains the range of a real random variable $$Y$$, then $$g\{E[Y|X]\} \le E[g(Y)|X]$$ a.s.
(CE15): Suppose $$E[|Y|^p] < \infty$$ and $$E[|Z|^p] < \infty$$ for $$1 \le p < \infty$$. Then $$E\{|E[Y|X] - E[Z|X]|^p\} \le E[|Y - Z|^p] < \infty$$

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