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 \(E\{|E[Y|X] - E[Z|X]|^p\} \le E[|Y - Z|^p] < \infty\)