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(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(X∈M)E[IM(X)]=P(X∈M) and E[IM(X)IN(Y)]=P(X∈M,Y∈N)E[IM(X)IN(Y)]=P(X∈M,Y∈N) 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.
- X≥0a.s.X≥0a.s. implies E[X]≥0E[X]≥0, with equality iff X=0a.s.X=0a.s.
- X≥Ya.s.X≥Ya.s. implies E[X]≥E[Y]E[X]≥E[Y], with equality iff X=Ya.s.X=Ya.s.
- (E4) : Fundamental lemma. If X≥0X≥0 is bounded, and
{Xn:1≤n}{Xn:1≤n} 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,0≤Xn≤Xn+1a.s . n,0≤Xn≤Xn+1a.s . and Xn→Xa.s . Xn→Xa.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 ≥
- 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.
- 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 Xn≥0a.s.Xn≥0a.s., for all n,
then E[lim infXn]≤lim infE[Xn]E[lim infXn]≤lim infE[Xn]
- (E7) : Dominated convergence. If real or complex Xn→Xa.s.Xn→Xa.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.
- If X is integrable over E, and E=⋁i=1∞EiE=⋁i=1∞Ei
(disjoint union),
then E[IEX]=∑i=1∞E[IEiX]E[IEX]=∑i=1∞E[IEiX]
- If ∑n=1∞E[|Xn|]<∞∑n=1∞E[|Xn|]<∞, then ∑n=1∞|Xn|<∞a.s.∑n=1∞|Xn|<∞a.s. and
E[∑n=1∞Xn]=∑n=1∞E[Xn]E[∑n=1∞Xn]=∑n=1∞E[Xn]
- (E9) :
Some integrability conditions
- X is integrable iff both X+ and X- are integrable
iff |X||X| is integrable.
- X is integrable iff E[I{|X|>a}|X|]→0E[I{|X|>a}|X|]→0 as a→∞a→∞
- If X is integrable, then X is a.s. finite
- 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 a≤X≤ba.s.a≤X≤ba.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 g≥0g≥0 and nondecreasing for t≥0t≥0
and a≥0a≥0, 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]|2≤E[|X|2]E[|Y|2]|E[XY]|2≤E[|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 1≤p,q1≤p,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/p≤E[|X|p]1/p+E[|Y|p]1/pE[|X+Y|p]1/p≤E[|X|p]1/p+E[|Y|p]1/p
(4) - (E18): Independence and expectation. The following conditions are equivalent.
- The pair {X,Y}{X,Y} is independent
- 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
- 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
- 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 X≥0X≥0 with E[F(X)]<∞E[F(X)]<∞. Then,
(1)E[F(X)]=∫0∞P(X≥t)F(dt)and(2)E[F*(X)]=∫0∞P(X>t)F(dt)(1)E[F(X)]=∫0∞P(X≥t)F(dt)and(2)E[F*(X)]=∫0∞P(X>t)F(dt)
(5) - If X is integrable, then E[X]=∫-∞∞[u(t)-FX(t)]dtE[X]=∫-∞∞[u(t)-FX(t)]dt
- If X,YX,Y are integrable, then E[X-Y]=∫-∞∞[FY(t)-FX(t)]dtE[X-Y]=∫-∞∞[FY(t)-FX(t)]dt
- If X≥0X≥0 is integrable, then
∑n=0∞P(X≥n+1)≤E[X]≤∑n=0∞P(X≥n)≤N∑k=0∞P(X≥kN),forallN≥1∑n=0∞P(X≥n+1)≤E[X]≤∑n=0∞P(X≥n)≤N∑k=0∞P(X≥kN),forallN≥1
(6) - If integrable X≥0X≥0 is integer-valued, then
E[X]=∑n=1∞P(X≥n)=∑n=0∞P(X>n)E[X]=∑n=1∞P(X≥n)=∑n=0∞P(X>n)E[X2]=∑n=1∞(2n-1)P(X≥n)=∑n=0∞(2n+1)P(X>n)E[X2]=∑n=1∞(2n-1)P(X≥n)=∑n=0∞(2n+1)P(X>n)
- 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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