# properties of expected value

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

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

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

###### Proof.

1) Let’s define

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

Then by hypothesis

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

and

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

We have:

 $\displaystyle E[X]$ $\displaystyle=$ $\displaystyle\int_{\Omega}X\left(\omega\right)dP$ $\displaystyle=$ $\displaystyle\int_{\Omega\backslash F}X\left(\omega\right)dP+\int_{F}X\left(% \omega\right)dP$ $\displaystyle=$ $\displaystyle\int_{F}X\left(\omega\right)dP$ $\displaystyle=$ $\displaystyle\int_{F}cdP$ $\displaystyle=$ $\displaystyle c\Pr\left\{F\right\}=c.$

2) [to be done].

3) Let’s define

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

Then by hypothesis

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

and

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

We have, keeping in mind property 2),

 $\displaystyle E[Y]-E[X]$ $\displaystyle=$ $\displaystyle E[Y-X]$ $\displaystyle=$ $\displaystyle\int_{\Omega}\left[Y\left(\omega\right)-X\left(\omega\right)% \right]dP$ $\displaystyle=$ $\displaystyle\int_{\Omega\backslash F}\left[Y\left(\omega\right)-X\left(\omega% \right)\right]dP+\int_{F}\left[Y\left(\omega\right)-X\left(\omega\right)\right% ]dP$ $\displaystyle=$ $\displaystyle\int_{F}\left[Y\left(\omega\right)-X\left(\omega\right)\right]dP% \geq 0.$

Title properties of expected value PropertiesOfExpectedValue 2013-03-22 16:16:05 2013-03-22 16:16:05 Andrea Ambrosio (7332) Andrea Ambrosio (7332) 6 Andrea Ambrosio (7332) Theorem msc 60-00