I got a stochastic problem and hope some of you can help me!
It's about stochastic variables $X,Y \in L^{\infty}(P)$, which have the same (finite) stochstic moments $\forall n \in \mathbb{N}: ~\mathbb{E}X^n = \mathbb{E}Y^n $
(a) First I need to show, that for an arbitrary integrable and continous function $f $ follows: $\mathbb{E}f(X) = \mathbb{E}f(Y)$
(b) Furthermore I should show, that the distributions $X$ and $Y$ even are identical.
I've already thought about the two problems. I guess the integrability is only necessary for definig/calculation $\mathbb{E}f(X)$, so this is not the propoerty, which is used in the proof. My approach for (a) was the Weierstraß - Theorem. The theorem claims, that $\forall f \in C([a,b])$ with $-\infty < a < b < \infty$ there holds: $\forall \epsilon >0 ~ \exists n \in \mathbb{N}: \exists p \in P_n: ||f-p ||_{\infty} < \epsilon$ on $[a,b]$.
So for a continous $f$ I can approximate it by $p \in P_n$: $f(z) \approx \sum_{k=0}^{n} a_k z^k$. So it follows $\mathbb{E}f(X) = \mathbb{E}\left[ \sum_{k=0}^{n} a_k z^k \right] = \sum_{k=0}^{n} a_k \mathbb{E}X^n = \sum_{k=0}^{n} a_k \mathbb{E}Y^n = \mathbb{E}f(Y)$.
At least that's the plan. The requirement for the Weierstraß-Theorem was a finite interval $[a,b]$, but according to the indication I cannot restrain to finite $a,b$. How can I generalize my result? (Or is there a general misstake in my approach?)
For (b) I dont't have an idea how to proof this. It might be possible, that I Need to apply a) but don't see how, due to I made a misstake at a)? I am a little bit confused, due to I thought $X,Y$ having same stochastical momenta is equal to $X$ idential to $Y$. But obviously this is exactly what we need to proof.
I hope some of you can me!
Thanks a lot! :)