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Appendix E to Applied Probability: Properties of mathematical expectation

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

Summary: A compilation of properties of the fundamental concept of mathematical expectation. Not all of these properties are used explicitly in this treatment, but they are included for reference.

E [ g ( X ) ] = g ( X ) d P E [ g ( X ) ] = g ( X ) d P
(1)

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 IM(X)IM(X) involves the tacit assumption that M is a Borel set on the codomain of X.

  • (E1) : E[aIA]=aP(A)E[aIA]=aP(A), any constant a, any event A
  • (E1a) : E[IM(X)]=P(XM)E[IM(X)]=P(XM) and E[IM(X)IN(Y)]=P(XM,YN)E[IM(X)IN(Y)]=P(XM,YN) for any Borel sets M,NM,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]a,b,E[aX+bY]=aE[X]+bE[Y] (Extends to any finite linear combination)
  • (E3) : Positivity; monotonicity.
    1. X0a.s.X0a.s. implies E[X]0E[X]0, with equality iff X=0a.s.X=0a.s.
    2. XYa.s.XYa.s. implies E[X]E[Y]E[X]E[Y], with equality iff X=Ya.s.X=Ya.s.
  • (E4) : Fundamental lemma. If X0X0 is bounded, and {Xn:1n}{Xn:1n} is a.s. nonnegative, nondecreasing, with limnXn(ω)X(ω)limnXn(ω)X(ω) for a.e. ω, then limnE[Xn]E[X]limnE[Xn]E[X]
  • (E4a): Monotone convergence. If for all n,0XnXn+1a.s . n,0XnXn+1a.s . and XnXa.s . XnXa.s . ,
    then E[Xn]E[X]E[Xn]E[X] (The theorem also holds if E[X]=E[X]=)

    *****
  • (E5) : Uniqueness. ** is to be read as one of the symbols ,=, or ,=, or
    1. E[IM(X)g(X)]*E[IM(X)h(X)]E[IM(X)g(X)]*E[IM(X)h(X)] for all M iff g(X)*h(X)a.s.g(X)*h(X)a.s.
    2. E[IM(X)IN(Z)g(X,Z)]=E[IM(X)IN(Z)h(X,Z)]E[IM(X)IN(Z)g(X,Z)]=E[IM(X)IN(Z)h(X,Z)] for all M,NM,N iff g(X,Z)=h(X,Z)a.s.g(X,Z)=h(X,Z)a.s.
  • (E6) : Fatou's lemma. If Xn0a.s.Xn0a.s., for all n, then E[lim infXn]lim infE[Xn]E[lim infXn]lim infE[Xn]
  • (E7) : Dominated convergence. If real or complex XnXa.s.XnXa.s., |Xn|Ya.s.|Xn|Ya.s. for all n, and Y is integrable, then limnE[Xn]=E[X]limnE[Xn]=E[X]
  • (E8) : Countable additivity and countable sums.
    1. If X is integrable over E, and E=i=1EiE=i=1Ei (disjoint union), then E[IEX]=i=1E[IEiX]E[IEX]=i=1E[IEiX]
    2. If n=1E[|Xn|]<n=1E[|Xn|]<, then n=1|Xn|<a.s.n=1|Xn|<a.s. and E[n=1Xn]=n=1E[Xn]E[n=1Xn]=n=1E[Xn]
  • (E9) : Some integrability conditions
    1. X is integrable iff both X+ and X- are integrable iff |X||X| is integrable.
    2. X is integrable iff E[I{|X|>a}|X|]0E[I{|X|>a}|X|]0 as aa
    3. If X is integrable, then X is a.s. finite
    4. If E[X]E[X] exists and P(A)=0P(A)=0, then E[IAX]=0E[IAX]=0
  • (E10): Triangle inequality. For integrable X, real or complex, |E[X]|E[|X|]|E[X]|E[|X|]
  • (E11): Mean-value theorem. If aXba.s.aXba.s. on A, then aP(A)E[IAX]bP(A)aP(A)E[IAX]bP(A)
  • (E12): For nonnegative, Borel g,E[g(X)]aP(g(X)a)g,E[g(X)]aP(g(X)a)
  • (E13): Markov's inequality. If g0g0 and nondecreasing for t0t0 and a0a0, then
    g(a)P(|X|a)E[g(|X|)]g(a)P(|X|a)E[g(|X|)]
    (2)
  • (E14): Jensen's inequality. If g is convex on an interval which contains the range of random variable X, then g(E[X])E[g(X)]g(E[X])E[g(X)]
  • (E15): Schwarz' inequality. For X,YX,Y real or complex, |E[XY]|2E[|X|2]E[|Y|2]|E[XY]|2E[|X|2]E[|Y|2], with equality iff there is a constant c such that X=cYa.s.X=cYa.s.
  • (E16): Hölder's inequality. For 1p,q1p,q, with 1p+1q=11p+1q=1, and X,YX,Y real or complex,
    E[|XY|]E[|X|p]1/pE[|Y|q]1/qE[|XY|]E[|X|p]1/pE[|Y|q]1/q
    (3)
  • (E17): Minkowski's inequality. For 1<p1<p and X,YX,Y real or complex,
    E[|X+Y|p]1/pE[|X|p]1/p+E[|Y|p]1/pE[|X+Y|p]1/pE[|X|p]1/p+E[|Y|p]1/p
    (4)
  • (E18): Independence and expectation. The following conditions are equivalent.
    1. The pair {X,Y}{X,Y} is independent
    2. E[IM(X)IN(Y)]=E[IM(X)]E[IN(Y)]E[IM(X)IN(Y)]=E[IM(X)]E[IN(Y)] for all Borel M,NM,N
    3. E[g(X)h(Y)]=E[g(X)]E[h(Y)]E[g(X)h(Y)]=E[g(X)]E[h(Y)] for all Borel g,hg,h such that g(X),h(Y)g(X),h(Y) are integrable.
  • (E19): Special case of the Radon-Nikodym theorem If g(Y)g(Y) is integrable and X is a random vector, then there exists a real-valued Borel function e(·)e(·), defined on the range of X, unique a.s. [PX][PX], such that E[IM(X)g(Y)]=E[IM(X)e(X)]E[IM(X)g(Y)]=E[IM(X)e(X)] for all Borel sets M on the codomain of X.
  • (E20): Some special forms of expectation
    1. Suppose F is nondecreasing, right-continuous on [0,)[0,), with F(0-)=0F(0-)=0. Let F*(t)=F(t-0)F*(t)=F(t-0). Consider X0X0 with E[F(X)]<E[F(X)]<. Then,
      (1)E[F(X)]=0P(Xt)F(dt)and(2)E[F*(X)]=0P(X>t)F(dt)(1)E[F(X)]=0P(Xt)F(dt)and(2)E[F*(X)]=0P(X>t)F(dt)
      (5)
    2. If X is integrable, then E[X]=-[u(t)-FX(t)]dtE[X]=-[u(t)-FX(t)]dt
    3. If X,YX,Y are integrable, then E[X-Y]=-[FY(t)-FX(t)]dtE[X-Y]=-[FY(t)-FX(t)]dt
    4. If X0X0 is integrable, then
      n=0P(Xn+1)E[X]n=0P(Xn)Nk=0P(XkN),forallN1n=0P(Xn+1)E[X]n=0P(Xn)Nk=0P(XkN),forallN1
      (6)
    5. If integrable X0X0 is integer-valued, then E[X]=n=1P(Xn)=n=0P(X>n)E[X]=n=1P(Xn)=n=0P(X>n)E[X2]=n=1(2n-1)P(Xn)=n=0(2n+1)P(X>n)E[X2]=n=1(2n-1)P(Xn)=n=0(2n+1)P(X>n)
    6. If Q is the quantile function for FX, then E[g(X)]=01g[Q(u)]duE[g(X)]=01g[Q(u)]du

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