properties of expected value


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

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

3) (monotonicity) Let X, Y be random variables such that Pr{XY}=1 and E[|X|]<, E[|Y|]<; then E[X]E[Y].

Proof.

1) Let’s define

F={ωΩ:X(ω)=c};

Then by hypothesisMathworldPlanetmath

Pr{Ω\F}=0

and

Pr{F}=1.

We have:

E[X] = ΩX(ω)𝑑P
= Ω\FX(ω)𝑑P+FX(ω)𝑑P
= FX(ω)𝑑P
= Fc𝑑P
= cPr{F}=c.

2) [to be done].

3) Let’s define

F={ωΩ:X(ω)Y(ω)};

Then by hypothesis

Pr{Ω\F}=0

and

Pr{F}=1.

We have, keeping in mind property 2),

E[Y]-E[X] = E[Y-X]
= Ω[Y(ω)-X(ω)]𝑑P
= Ω\F[Y(ω)-X(ω)]𝑑P+F[Y(ω)-X(ω)]𝑑P
= F[Y(ω)-X(ω)]𝑑P0.

Title properties of expected value
Canonical name PropertiesOfExpectedValue
Date of creation 2013-03-22 16:16:05
Last modified on 2013-03-22 16:16:05
Owner Andrea Ambrosio (7332)
Last modified by Andrea Ambrosio (7332)
Numerical id 6
Author Andrea Ambrosio (7332)
Entry type Theorem
Classification msc 60-00