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

Module by: Paul E Pfeiffer. E-mail the author

Summary: A compilation of key properties of this concept, which plays a central role in many areas of theoretical and applied probability. Many topics which are often approached intuitively and informally can be given precise formulation and analysis.

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]e(X)=E[g(Y)|X] a.s. iff E[IM(X)g(Y)]=E[IM(X)e(X)]E[IM(X)g(Y)]=E[IM(X)e(X)] for each Borel set M on the codomain of X.
  • (CE1a): If P(XM)>0P(XM)>0, then E[IM(X)e(X)]=E[g(Y)|XM]P(XM)E[IM(X)e(X)]=E[g(Y)|XM]P(XM)
  • (CE1b): Law of total probability. E[g(Y)]=E{E[g(Y)|X]}E[g(Y)]=E{E[g(Y)|X]}
  • (CE2): Linearity. For any constants a,ba,b
    E [ a g ( Y ) + b h ( Z ) | X ] = a E [ g ( Y ) | X ] + b E [ h ( Z ) | X ] a . s . E [ a g ( Y ) + b h ( Z ) | X ] = a E [ g ( Y ) | X ] + b E [ h ( Z ) | X ] a . s .
    (Extends to any finite linear combination)
  • (CE3): Positivity; monotonicity.
    1. g(Y)0g(Y)0 a.s. implies E[g(Y)|X]0E[g(Y)|X]0 a.s.
    2. g(Y)h(Z)g(Y)h(Z) a.s. implies E[g(Y)|X]E[h(Z)|X]E[g(Y)|X]E[h(Z)|X] a.s.
  • (CE4): Monotone convergence. YnYYnY a.s. monotonically implies E [ Y n | X ] E [ Y | X ] a . s . E [ Y n | X ] E [ Y | X ] a . s .
  • (CE5): Independence. {X,Y}{X,Y} is an independent pair
    • iff E[g(Y)|X]=E[g(Y)]E[g(Y)|X]=E[g(Y)] a.s. for all Borel functions g
    • iff E[IN(Y)|X]=E[IN(Y)]E[IN(Y)|X]=E[IN(Y)] a.s. for all Borel sets N on the codomain of Y
  • (CE6): e(X)=E[g(Y)|X]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 E [ h ( X ) g ( Y ) ] = E [ h ( X ) e ( X ) ] a . s . for any Borel function h
  • (CE7): E[h(X)|X]=h(X)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]E[h(X)g(Y)|X]=h(X)E[g(Y)|X] a.s. for any Borel function h
  • (CE9): If X=h(W)X=h(W), then E{E[g(Y)|X]|W}=E{E[g(Y)|W]|X}=E[g(Y)|X]E{E[g(Y)|X]|W}=E{E[g(Y)|W]|X}=E[g(Y)|X], a.s.
  • (CE9a): E{E[g(Y)|X]|X,Z}=E{E[g(Y)|X,Z]|X}=E[g(Y)|X]E{E[g(Y)|X]|X,Z}=E{E[g(Y)|X,Z]|X}=E[g(Y)|X] a.s.
  • (CE9b): If X=h(W)X=h(W) and W=k(X)W=k(X), with h,kh,k Borel functions, then E [ g ( Y ) | X ] = E [ g ( Y ) | W ] a . s . E [ g ( Y ) | X ] = E [ g ( Y ) | W ] a . s .
  • (CE10): If g is a Borel function such that E[g(t,Y)]E[g(t,Y)] is finite for all t on the range of X and E[g(X,Y)]E[g(X,Y)] is finite, then
    1. E[g(X,Y)|X=t]=E[g(t,Y)|X=t]E[g(X,Y)|X=t]=E[g(t,Y)|X=t] a.s. [PX][PX]
    2. If {X,Y}{X,Y} is independent, then E[g(X,Y)|X=t]=E[g(t,Y)]E[g(X,Y)|X=t]=E[g(t,Y)] a.s. [PX][PX]
  • (CE11): Suppose {X(t):tT}{X(t):tT} 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)X(S) is a random variable.
    If E[X(t)]E[X(t)] is finite for all t in T and E[X(S)]E[X(S)] is finite, then
    1. E[X(S)|S=t]=E[X(t)|S=t]E[X(S)|S=t]=E[X(t)|S=t] a.s. [PS][PS]
    2. If, in addition, {S,XT}{S,XT} is independent, then E[X(S)|S=t]=E[X(t)]E[X(S)|S=t]=E[X(t)] a.s. [PS][PS]
  • (CE12): Countable additivity and countable sums.
    1. If Y is integrable on A and A=n=1AnA=n=1An,
      then E[IAY|X]=n=1E[IAnY|X]E[IAY|X]=n=1E[IAnY|X] a.s.
    2. If n=1E[|Yn|]<n=1E[|Yn|]< , then En=1Yn|X=n=1E[Yn|X]En=1Yn|X=n=1E[Yn|X] a.s.
  • (CE13): Triangle inequality. |E[g(Y)|X]|E[|g(Y)||X]|E[g(Y)|X]|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 ] } E [ g ( Y ) | X ] a . s . g { E [ Y | X ] } E [ g ( Y ) | X ] a . s .
  • (CE15): Suppose E[|Y|p]<E[|Y|p]< and E[|Z|p]<E[|Z|p]< for 1p<1p<. Then E { | E [ Y | X ] - E [ Z | X ] | p } E [ | Y - Z | p ] < E { | E [ Y | X ] - E [ Z | X ] | p } E [ | Y - Z | p ] <

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