17.4 Appendix e to applied probability: properties of mathematical

We suppose, without repeated assertion, that the random variables and Borel functions of random variables or random vectors are integrable. Useof an expression such as $I_M\left(X\right)$ involves the tacit assumption that M is a Borel set on the codomain of X .

• (E1) $E\left[a{I}_A\right]=aP\left(A\right)$ , any constant a , any event A
• (E1a) $E\left[I_M\left(X\right)\right]=P(X\in M)$ and $E\left[I_M\left(X\right)I_N\left(Y\right)\right]=P(X\in M,Y\in N)$ for any Borel sets $M,N$ (Extends to any finite product of such indicator functions of random vectors)
• (E2) Linearity . For any constants $a,b,E[aX+bY]=aE\left[X\right]+bE\left[Y\right]$ (Extends to any finite linear combination)
• (E3) Positivity; monotonicity .
  1. $X\ge 0\mathrm{a}.\mathrm{s}.$ implies $E\left[X\right]\ge 0$ , with equality iff $X=0\mathrm{a}.\mathrm{s}.$
  2. $X\ge Y\mathrm{a}.\mathrm{s}.$ implies $E\left[X\right]\ge E\left[Y\right]$ , with equality iff $X=Y\mathrm{a}.\mathrm{s}.$
• (E4) Fundamental lemma . If $X\ge 0$ is bounded, and $\{X_n:1\le n\}$ is a.s. nonnegative, nondecreasing, with ${lim}_n{X}_n\left(\omega \right)\ge X\left(\omega \right)$ for a.e. ω , then ${lim}_{n}E\left[X_n\right]\ge E\left[X\right]$
• (E4a) Monotone convergence . If for all $n,0\le X_n\le X_{n+1}\mathrm{a}.\mathrm{s}.$ and $X_n\to X\mathrm{a}.\mathrm{s}.$ ,
  then $E\left[X_n\right]\to E\left[X\right]$ (The theorem also holds if $E\left[X\right]=\infty$ )
  
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• (E5) Uniqueness . $*$ is to be read as one of the symbols $\le ,=,\text{or}\ge$
  1. $E\left[I_M\left(X\right)g\left(X\right)\right]*E\left[I_M\left(X\right)h\left(X\right)\right]$ for all M iff $g\left(X\right)*h\left(X\right)\mathrm{a}.\mathrm{s}.$
  2. $E\left[I_M\left(X\right)I_N\left(Z\right)g(X,Z)\right]=E\left[I_M\left(X\right)I_N\left(Z\right)h(X,Z)\right]$ for all $M,N$ iff $g(X,Z)=h(X,Z)\mathrm{a}.\mathrm{s}.$
• (E6) Fatou's lemma . If $X_n\ge 0\mathrm{a}.\mathrm{s}.$ , for all n , then $E[lim\; inf{X}_n]\le lim\; infE\left[X_n\right]$
• (E7) Dominated convergence . If real or complex $X_n\to X\mathrm{a}.\mathrm{s}.$ , $|X_n|\le Y\mathrm{a}.\mathrm{s}.$ for all n , and Y is integrable, then ${lim}_{n}E\left[X_n\right]=E\left[X\right]$
• (E8) Countable additivity and countable sums .
  1. If X is integrable over E , and ${\displaystyle E=\underset{i=1}{\overset{\infty }{\bigvee }}{E}_i}$ (disjoint union), then ${\displaystyle E\left[I_{E}X\right]=\sum _{i=1}^{\infty }E\left[I_{E_i}X\right]}$
  2. If ${\displaystyle \sum _{n=1}^{\infty }E\left[\right|X_n\left|\right]<\infty }$ , then ${\displaystyle \sum _{n=1}^{\infty }\left|X_n\right|<\infty \mathrm{a}.\mathrm{s}.}$ and ${\displaystyle E\left[\sum _{n=1}^{\infty }{X}_n\right]=\sum _{n=1}^{\infty }E\left[X_n\right]}$
• (E9) Some integrability conditions
  1. X is integrable iff both X ⁺ and X ^- are integrable iff $\left|X\right|$ is integrable.
  2. X is integrable iff $E\left[I_{\left\{\right|X|>a\}}\right|X\left|\right]\to 0$ as $a\to \infty$
  3. If X is integrable, then X is a.s. finite
  4. If $E\left[X\right]$ exists and $P\left(A\right)=0$ , then $E\left[I_{A}X\right]=0$
• (E10) Triangle inequality . For integrable X , real or complex, $\left|E\right[X\left]\right|\le E\left[\right|X\left|\right]$
• (E11) Mean-value theorem . If $a\le X\le b\mathrm{a}.\mathrm{s}.$ on A , then $aP\left(A\right)\le E\left[I_{A}X\right]\le bP\left(A\right)$
• (E12) For nonnegative, Borel $g,E\left[g\right(X\left)\right]\ge aP\left(g\right(X)\ge a)$
• (E13) Markov's inequality . If $g\ge 0$ and nondecreasing for $t\ge 0$ and $a\ge 0$ , then
  $$g\left(a\right)P\left(\right|X|\ge a)\le E\left[g\right(\left|X\right|\left)\right]$$
• (E14) Jensen's inequality . If g is convex on an interval which contains the range of random variable X , then $g\left(E\right[X\left]\right)\le E\left[g\right(X\left)\right]$
• (E15) Schwarz' inequality . For $X,Y$ real or complex, ${\left|E\left[XY\right]\right|}^2\le {E\left[\right|X|}^2{\left]E\right[\left|Y\right|}^2]$ , with equality iff there is a constant c such that $X=cY\mathrm{a}.\mathrm{s}.$
• (E16) Hölder's inequality . For $1\le p,q$ , with ${\displaystyle \frac{1}{p}+\frac{1}{q}=1}$ , and $X,Y$ real or complex,
  $$E\left[\right|XY\left|\right]\le {E\left[\right|X|}^p{]}^{1/p}{E\left[\right|Y|}^q{]}^{1/q}$$
• (E17) Minkowski's inequality . For $1<p$ and $X,Y$ real or complex,
  $${E\left[\right|X+Y|}^p{]}^{1/p}\le {E\left[\right|X|}^p{]}^{1/p}+{E\left[\right|Y|}^p{]}^{1/p}$$
• (E18) Independence and expectation . The following conditions are equivalent.
  1. The pair $\{X,Y\}$ is independent
  2. $E\left[I_M\left(X\right)I_N\left(Y\right)\right]=E\left[I_M\left(X\right)\right]E\left[I_N\left(Y\right)\right]$ for all Borel $M,N$
  3. $E\left[g\right(X\left)h\right(Y\left)\right]=E\left[g\right(X\left)\right]E\left[h\right(Y\left)\right]$ for all Borel $g,h$ such that $g\left(X\right),h\left(Y\right)$ are integrable.
• (E19) Special case of the Radon-Nikodym theorem If $g\left(Y\right)$ is integrable and X is a random vector, then there exists a real-valued Borel function $e(·)$ , defined on the range of X , unique a.s. $\left[P_X\right]$ , such that $E\left[I_M\left(X\right)g\left(Y\right)\right]=E\left[I_M\left(X\right)e\left(X\right)\right]$ for all Borel sets M on the codomain of X .
• (E20) Some special forms of expectation
  1. Suppose F is nondecreasing, right-continuous on $[0,\infty )$ , with $F\left(0^{-}\right)=0$ . Let $F^{*}\left(t\right)=F(t-0)$ . Consider $X\ge 0$ with $E\left[F\right(X\left)\right]<\infty$ . Then,
     $$\left(1\right)E\left[F\left(X\right)\right]={\int }_0^{\infty }P(X\ge t)F\left(dt\right)\text{and}\left(2\right)E\left[F^{*}\left(X\right)\right]={\int }_0^{\infty }P(X>t)F\left(dt\right)$$
  2. If X is integrable, then $E\left[X\right]={\int }_{-\infty }^{\infty }[u\left(t\right)-F_X\left(t\right)]dt$
  3. If $X,Y$ are integrable, then $E[X-Y]={\int }_{-\infty }^{\infty }[F_Y\left(t\right)-F_X\left(t\right)]dt$
  4. If $X\ge 0$ is integrable, then
     $$\sum _{n=0}^{\infty }P(X\ge n+1)\le E\left[X\right]\le \sum _{n=0}^{\infty }P(X\ge n)\le N\sum _{k=0}^{\infty }P(X\ge kN),\text{for}\text{all}N\ge 1$$
  5. If integrable $X\ge 0$ is integer-valued, then ${\displaystyle E\left[X\right]=\sum _{n=1}^{\infty }P(X\ge n)=\sum _{n=0}^{\infty }P(X>n)}$ ${\displaystyle E\left[X^2\right]=\sum _{n=1}^{\infty }(2n-1)P(X\ge n)=\sum _{n=0}^{\infty }(2n+1)P(X>n)}$
  6. If Q is the quantile function for F _X , then ${\displaystyle E\left[g\left(X\right)\right]={\int }_0^{1}g\left[Q\left(u\right)\right]du}$