Expected Value of Random Variables

#ProbabilityTheory

Subjects: Probability Theory
Links: Random Variables, Probability Functions for Random Variables, Riemann-Steiltjes Integral on R, Measurable Functions

Expected Value

Let $X$ be a random variable with a distribution function $F$ . The expected value of $X$ , denoted as $E [X]$ , it is defined as the number

E [X] := \int_{R} x d F (x)

when this integral is absolutely convergent, i.e., when the integral $\int_{R} | x | d F (x) < \infty$ converges, in this case we say that $X$ is integrable or that it has finite expected value

Let $X$ be discrete random variable with probability mass function $f$ . The expected value of $X$ is defined to be

E [X] = \sum_{x} x f (x)

supposing that the sum is absolutely convergent, meaning, when the sum of the values converges.
In the particular cases:
Let $X$ be a absolutely continuous random variable with probability density function $f$ , then the expected value is

E [X] = \int_{R} x f (x) d x

Let $X$ be a random variable with a distribution function $F_{X}$ , and let $g : R \to R$ be a Borel measurable function, then $g (X)$ is a random variable, and we if try to calculate its expected value we get:

E [g (X)] = \int_{R} x d F_{g (X)} (x)

Law of the unconscious statistician

Let $X$ be a random variable with a distribution function $F_{X}$ , and let $g : R \to R$ be a Borel measurable function, such that the random variable $g (X)$ has finite expected value. Then

E [g (X)] = \int_{R} g (x) d F_{X} (x)

In the particular cases that:
Let $X$ be continuous random variable with probability density function $f$ , and $g : R \to R$ be a function such that $g (X)$ is a random variable with finite expected value then

E [g (X)] = \sum_{x} g (x) f (x)

Let $X$ be discrete random variable with probability mass function $f$ , and $g : R \to R$ be a function such that $g (X)$ is a random variable with finite expected value then

E [g (X)] = \int_{R} g (x) f (x) d x

Let $(X, Y)$ be a random vector $ϕ : R^{2} \to R$ be Borel measurable such that $ϕ (X, Y)$ be a random variable with finite expected value. Then we define it as

E [ϕ (X, Y)] = \int_{R} x d F_{ϕ (X, Y)} (x)

Expected Value of a Function of a Random Vector

Let $(X, Y)$ be a random vector $ϕ : R^{2} \to R$ be Borel measurable such that $ϕ (X, Y)$ be a random variable with finite expected value. Then we define it as

E [ϕ (X, Y)] = \int_{R^{2}} ϕ (x, y) d F_{X, Y} (x, y)

using Riemann-Stieltjes Integral in Rn. In the special case where $X$ and $Y$ are independent. We can actually simplify it to two integrals

E [ϕ (X, Y)] = \int_{R^{2}} ϕ (x, y) d F_{X} (x) d F_{Y} (y)

Prop: Let $X$ and $Y$ be continuous random variables defined over the same probability space with a conjoined probability density function $f (x, y)$ . Let $g : R^{2} \to R$ a function such that $g (X, Y)$ is a random variable with finite expected value. Then

E [g (X, Y)] = \int_{R} \int_{R} g (x, y) f (x, y) d y d x

If the random variables are discrete with a conjoined probability mass function $f (x, y)$ . Let $g : R^{2} \to R$ a function such that $g (X, Y)$ is a random variable with finite expected value. Then

E [g (X, Y)] = \sum_{x} \sum_{y} g (x, y) f (x, y)

Prop: Properties of the expecte value. Let $X$ and $Y$ be random variables with finite expected value and $c$ a constant. Then

$E [c] = c$
$E [c X] = c E [X]$
If $X \geq 0$ , then $E [X] \geq 0$
$E [X + Y] = E [X] + E [Y]$

Then we know that the expected value behaves linearly.

Let $X$ be a random variable with a distribution function $F$ , that admits a decomposition:

F (x) = α F^{d} (x) + (1 - α) F^{c} (x) $ $ w i t h $ 0 \leq α \leq 1 $, w h e r e $ F^{d} $ i s a d i s c r e t e d i s t r i b u t i o n f u n c t i o n a n d $ F^{d} $ i s a c o n t i n u o u s o n e . L e t $ X_{d} $ h a v e t h e d i s t r i b u t i o n $ F^{d} $ a n d $ X_{c} $ h a v e t h e d i s t r i b u t i o n $ F^{c} $ . T h e n, $ X $ h a s a f i n i t e e x p e c t e d v a l u e, i f f, $ X_{d} $ a n d $ X_{c} $ h a v e f i n i t e e x p e c t e d v a l u e, a n d i n t h a t c a s e

E[X] = \alpha E[X_d]+ (1-\alpha)E[X_c]

* * * * * * * * * * * * P r o p : * * * * * * * * * * * * L e t $ X $ a n d $ Y $ b e i n d e p e n d e n t a n d b o t h w i t h f i n i t e e x p e c t e d v a l u e s, a n d l e t $ g, h : R^{2} \to R $ b e B o r e l m e a s u r a b l e a n d s u c h t h a t $ g (X) $ a n d $ h (Y) $ h a v e a f i n i t e e x p e c t e d v a l u e . T h e n

E[g(X)h(Y)] = E[g(X)]E[h(Y)]

i n p a r t i c u l a r,

E[XY] = E[X]E[Y]

* * * * * * * * * * * * P r o p : * * * * * * * * * * * * L e t $ X $ b e a d i s c r e t e r a n d o m v a r i a b l e w i t h c u m u l a t i v e p r o b a b i l i t y f u n c t i o n $ F (x) $, w i t h a f i n i t e e x p e c t e d v a l u e a n d p o s s i b l e v a l u e s i n t h e s e t $ N $ . T h e n

E[X] = \sum_{x \in \Bbb N}(1-F(x))

S i m i l a r l y, l e t $ X $ b e a c o n t i n u o u s r a n d o m v a r i a b l e w i t h a c u m u l a t i v e p r o b a b i l i t y f u n c t i o n $ F $, w i t h f i n i t e e x p e c t e d v a l u e a n d v a l u e s i n t h e i n t e r v a l $ [0, \infty) $ . T h e n

E[X] = \int_0^\infty (1-F(x)) , dx