I'm trying to prove the following: $$M_{X}(t)=\prod\limits_{i=1}^n M_{X_i}\left(\frac{t}{n}\right)$$ Where $X := \frac{1}{n} \sum\limits_{i=1}^n X_i $ and all the random variables are independent.
All I need to finish the prove is to show that $e^{\frac{t}{n}X_i}$ are independent which I'm not sure how to do. I was thinking in general if it is true that the bijection of independent random variables are also independent i.e. if we biject $X_i$ to $Y_i$, are the $Y_i$ independent given that the $X_i$ are?
Regards, Raxel.
One can define two random variables $X$ and $Y$ to be independent if (and only if) the following holds $$ E [u(X) v(Y)] = E[u(X)]\; E[v(Y)] \quad (*) $$ for any (measurable) function $u$ and $v$. Note that a special case of this is when $u(x) = 1\{ x \in A\}$ and $v(y) = \{y \in B\}$ for (meas.) sets $A$ and $B$. That is, if $u$ and $v$ are indicator functions. Then, the above reduces to $$P( X \in A, Y \in B) = P(X \in A) P(Y \in B)$$ which might be the definition that you have seen. (Though it may seem that this is a special case, it is in fact equivalent to the above mentioned ($*$))
Assuming ($*$), it is easy to see that if $X$ and $Y$ are independent, then $g(X)$ and $h(Y)$ are also independent (that is, they too satisfy ($*$)), since the composition of measurable functions is again a measurable function.