Probability Theory

>>8541262
>this autistic circlejerk
>probability theory
>wondering why he doesn't understand anything
lmao

>>8541281
This thread is a no-bully zone.

because the sigma algebra doesn't give any information about the random variable.

why would it change anything?

>>8541440
No, but the interesting thing here is that the expectation isn't a constant but a random variable again. Don't you think that's strange?

>>8541467
One look at the Wikipedia article where they discuss possible definitions makes it uninteresting, so no

>>8541537
Yes but I need a measure theoretic proof, please I am too retard to derive it myself.

>>8541262
E[X|F] is defined as an F-measurable random variable Y such that

[math]\int_A Y\, dP = \int_A X\, dP[/math] for all A in F. Clearly X satisfies this.

>>8541861
Wait, it's defined that way?

>>8541869
That's the definition in Dudley's Real Analysis and Probability.

>>8541467
I believe the definition of E[X|F], where F is a sigma-algebra is a generalization of E[X|Y] where Y is a random variable. E[X|Y] should be a random variable since the given information, i.e. Y, is random and not fixed.

that photo sure gave me a boner anon ;)

>>8541262
First, if [math]\mathcal{F'} \subseteq \mathcal{F}[/math] is a [math]\sigma[/math]-subalgebra, then we have an inclusion of Banach spaces [math]L^p(\Omega, \mathcal{F'}) \subseteq L^p(\Omega, \mathcal{F})[/math], since any function that is [math]\mathcal{F'}[/math]-measurable is certainly [math]\mathcal{F}[/math]-measurable. The random variable [math]X|\mathcal{F'}[/math] for [math]X \in L^p(\Omega, \mathcal{F})[/math] is the [math]\|\cdot\|_p[/math]-closest element of [math]L^p(\Omega,\mathcal{F'})[/math] to [math]X[/math], viewed as a linear subspace. In the [math]p=2[/math] case this is just the orthogonal projection. In the case [math]\mathcal{F'}=\mathcal{F}[/math], this just says that the closest element to [math]X[/math] in [math]\mathcal{F}[/math] is [math]X[/math].

>>8541467
replace [math]\mathcal{F}[/math] by the [math]\sigma[/math]-algebra [math]\{\emptyset, \Omega\}[/math]. The [math]X[/math] is just a constant.

>>8541971
Where can I read more about this? I'm guessing I have to google on "measure theoretic probability" or something, right? Do you specific book recommendations?

>>8541971
why would you ever make this definition for exponents other than p=2?

OP here, for anyone else interested, I've got a nice intuitive explanation here:

http://math.stackexchange.com/questions/690531/intuition-for-random-variable-being-sigma-algebra-measurable

>>8541467
oh you're right, that's fucked up!

>>8541898
but a sigma algebra isn't a random variable though, it's just the allowed subsets of omega.

Let's say X is the number of dots on the role of a fair dice. so Omega is {1,2,3,4,5,6}.

then filtration is just the set of all possible outcomes.

So the expectation of X given the filtration is simply the expectation of X (since the filtration tells us absolutely nothing about X we did not already know) which is equal to 3.5


So NO
E[X|F] is NOT always a random variable

Bad maths DISPROVEN by counter example

>>8543742
But I think that's wrong. [math] \mathbb{E} [ X | \mathcal{F} ] [/math] is a [math]\mathcal{F}[/math] -measurable function again.

Recall that [math] X [/math] is measurable, so if we have [math] \omega \in \Omega [/math], given the sigma algebra, we know which value X takes. So then:

[eqn] \mathbb{E} [ X | \mathcal{F} ](\omega) = X(\omega) [/eqn]

>>8543742
Here's a tip: don't post an answer if you have no idea what you're talking about

>>8543079
It kinda makes sense for L1, because we don't always have to assume second moment exists.

>>8543742
Even if your calculation was right (protip: it's not), a constant would still be a random variable, as it is a measurable map.

>>8544758
yeah that's true, but if you define the conditional expectation that way (i.e. as L1 projection), then you don't get the property [math] \int_A E(X\mid F)dP=\int_A XdP[/math] (in fact, this property is equivalent to saying that E(X| F) is the L2 projection of X onto the corresponding subspace)

the most common approach is to first define it as L2 projection for square-integrable X, and then extend this to all integrable X by approximating X with a sequence of L2 random variables