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 variables 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{X≤Y}=1 and E[|X|]<∞, E[|Y|]<∞; then E[X]≤E[Y].
Proof.
1) Let’s define
F={ω∈Ω:X(ω)=c}; |
Then by hypothesis
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={ω∈Ω: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)]𝑑P≥0. |
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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 |