# Conditioned expectancy value

caused expectancy values and caused probabilities concerning an part σ algebra represents a Verallgemeinerung of conditioned probabilities . They are used among other things with the formulation of Martingalen.

the density of a regular conditioned distribution $P (X \ in \, \ cdot \, \, | Y)< /math> in the faktorisierten form. ## arithmetic rules the equations are, as far as nothing else is indicated to understand in each case in such a way that the left side exists exactly (in the sense of the above definition) if the right side exists. • For trivial σ-algebra [itex] \ mathcal {B} = \ {\ emptyset, \ omega \}< /math> result simple expectancy values and probabilities: [itex] E (X|\ mathcal {B}) (\ omega) = E (X)$ for all $\ omega \ in \ omega< /math> [itex] P (A|\ mathcal {B}) (\ omega) = P (A)$ for all $\ omega \ in \ omega< /math> • Is [itex] X< /math> independently of [itex] \ mathcal {B}< /math>, then applies [itex] E (X|\ mathcal {B}) = E (X)$ nearly everywhere.
• Math <\> mathcal {B is} = \ mathcal {A}< /math> or $X< /math> measurably relative [itex] \ mathcal {B}< /math>, then applies [itex] E (X|\ mathcal {B}) = X$ nearly everywhere.
• For part σ algebras $\ mathcal {C} \ subset \ mathcal {B} \ subset \ mathcal {A}< /math> math <E> (E (X applies|\ mathcal {B})|\ mathcal {C}) = E (X|\ mathcal {C})< /math> and [itex] E (E (X|\ mathcal {C})|\ mathcal {B}) = E (X|\ mathcal {C})< /math> nearly everywhere. • It applies [itex] E (X_1 + X_2 | \ mathcal {B}) = E (X_1 | \ mathcal {B}) + E (X_2 | \ mathcal {B})$ nearly everywhere, if $X_1< /math> or [itex] X_2< /math> a finite expectancy value possesses. • It applies [itex] E (A X | \ mathcal {B}) = A E (X | \ mathcal {B})$ nearly everywhere for real numbers $A \ ne 0$.
• Monotonie: Out $X_1 \ le X_2$ follows $E (X_1 | \ mathcal {B}) \ le E (X_2 | \ mathcal {B})$ nearly everywhere, if the conditioned expectancy values exist.
• Monotonous convergence: Out $X_n \ uparrow X$ follows $E (X_n | \ mathcal {B}) \ uparrow E (X | \ mathcal {B})$ nearly everywhere, if the conditioned expectancy values exist and $E (X_1 | \ mathcal {B}) > - \ infty$ nearly everywhere.
• Jensen inequation: Is $f: \ mathbb {R} \ rightarrow \ mathbb {R}$ a convex function, then applies $f (E (X|\ mathcal {B})) \ le E (f (X)|\ mathcal {B})$ nearly everywhere, if the conditioned expectancy values exist.
• Is $Y$ measurably relative $\ mathcal {B}< /math>, then is [itex] E (YX|\ mathcal {B}) = Y E (X|\ mathcal {B})$ nearly everywhere, if the conditioned expectancy values exist. In particular is $E (Y (X - E (X|\ mathcal {B})) = 0$ nearly everywhere, D. h. the conditioned expectancy value $E (X|\ mathcal {B})$ the orthogonale projection of math X /math is in the sense of the dot product of L 2 <(>P<)> on the area [itex] \ mathcal {of B}< /math> - measurable functions.