I've this problem of statistics that I can't resolve. So I hope that someone can halp me.
The problem is this:
Let sampling moments $\overline{X_n^k}=\frac{1}{n}\sum_{i=1}^nX_i^k$, where for $k=1$ $\overline{X_n^k}=\overline{X_n}$.
Show that $E(\overline{X_n^k})=E(X)$, where $E(Y)$ is the mean value of $Y$.
I did this:
$E(\overline{X_n^k})=E(\frac{1}{n}\sum_{i=1}^nX_i^k)=\frac{1}{n}E(\sum_{i=1}^nX_i^k)$
I think that for ID of $X_i$, I can write:
$\frac{1}{n}E(\sum_{i=1}^nX_i^k)=\frac{1}{n}*nE(X_i^k)=E(X_i^k)$.
Is this correct? And how can I go on? I thought to use the moment generating functions, but I don't know use it.
Thank you for your help.