properties of expected value

1) (normalization) Let $X$ be almost surely constant randomvariable, i.e. $\mathrm{Pr}\left\{X=c\right\}=1$; then $E\left[X\right]=c$.

2) (linearity) Let $X$, $Y$ be random variables such that and and let $a$, $b$ be real numbers; then and $E\left[aX+bY\right]=aE\left[X\right]+bE\left[Y\right]$.

3) (monotonicity) Let $X$, $Y$ be random variables such that$\mathrm{Pr}\left\{X\le Y\right\}=1$ and , ; then $E\left[X\right]\le E\left[Y\right]$.

Proof.

1) Let’s define

$$F=\{\omega \in \mathrm{\Omega }:X\left(\omega \right)=c\};$$

Then by hypothesis

$$\mathrm{Pr}\left\{\mathrm{\Omega }\backslash \backslash F\right\}=0$$

and

$$\mathrm{Pr}\left\{F\right\}=1.$$

We have:

$E[X]$$=$${\displaystyle {\int }_{\mathrm{\Omega }}}X\left(\omega \right)𝑑P$

$=$${\displaystyle {\int }_{\mathrm{\Omega }\backslash \backslash F}}X\left(\omega \right)𝑑P+{\displaystyle {\int }_F}X\left(\omega \right)𝑑P$

$=$${\displaystyle {\int }_F}X\left(\omega \right)𝑑P$

$=$${\displaystyle {\int }_F}c𝑑P$

$=$$c\mathrm{Pr}\left\{F\right\}=c.$

2) [to be done].

3) Let’s define

$$F=\{\omega \in \mathrm{\Omega }:X\left(\omega \right)\le Y\left(\omega \right)\};$$

Then by hypothesis

$$\mathrm{Pr}\left\{\mathrm{\Omega }\backslash \backslash F\right\}=0$$

and

$$\mathrm{Pr}\left\{F\right\}=1.$$

We have, keeping in mind property 2),

$E[Y]-E[X]$$=$$E[Y-X]$

$=$${\displaystyle {\int }_{\mathrm{\Omega }}}\left[Y\left(\omega \right)-X\left(\omega \right)\right]𝑑P$

$=$${\displaystyle {\int }_{\mathrm{\Omega }\backslash \backslash F}}\left[Y\left(\omega \right)-X\left(\omega \right)\right]𝑑P+{\displaystyle {\int }_F}\left[Y\left(\omega \right)-X\left(\omega \right)\right]𝑑P$

$=$${\displaystyle {\int }_F}\left[Y\left(\omega \right)-X\left(\omega \right)\right]𝑑P\ge 0.$

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