Showing two random variables independent despite seemingly looking dependent I just met this in probability and it got me completely stumped:


  We define an i.i.d sequence of normally distributed random variables
  $ \{ X_n \}_{n=1}^{\infty} $ such that $ X_n \sim \mathcal{N} (0,\sigma ^2) $ and we define a random variable $ N \sim Pois(\Lambda) $ independent of the $ X_n $'s
  
  We are asked to show that $ X_N $ and N are independent random variables


I cannot seem to get past the simple fact that these seem clearly dependent as N clearly appears as a subscript for the X random variable so I do not really know what is wrong or how to prove that these are independent random variables so this is where I need help. Thanks all.
Independence follows from the fact that for every fixed $k\in\mathbb N$:$$P(X_N\in A)=\sum_{n=0}^{\infty} P(X_N\in A\mid N=n)P(N=n)=$$$$\sum_{n=0}^{\infty} P(X_n\in A)P(N=n)=\sum_{n=0}^{\infty} P(X_k\in A)P(N=n)=$$$$P(X_k\in A)=P(X_N\in A\mid N=k)$$
This is not different from rolling a green and a red die ans using a coin flip to decide which of the dice you use and which you ignore. You still have a coin flip and a die.



All that matters is that the $X_n$ are identically distributed (they need not be normally distributed and not independent of one anaother) and that $N$ is independent of all $(X_1,X_2,\ldots)$.

For any measurable $A$ and $n\in\Bbb N$, we have (letting $Y=X_N$ for clarity)
$$\begin{align} P(Y\in A\mid N=n)&amp;=P(X_n\in A\mid N=n)&amp;\text{def of $Y$}\\&amp;=\frac{P(X_n\in A\wedge N=n)}{P(N=n)}&amp;\text{def of conditional prob.}\\&amp;=P(X_n\in A)&amp;\text{independent}\\&amp;=P(X_1\in A)&amp;\text{identically distributed}\end{align}$$
so that (as one might have guessed)
$$ \begin{align}P(Y\in A)&amp;=\sum_n P(Y\in A\mid N=n)P(N=n)\\&amp;=\sum_nP(X_1\in A)P(N=n)\\&amp;=P(X_1\in A)\sum_nP(N=n)\\&amp;=P(X_1\in A)\end{align}$$
and hence 
$$P(Y\in A\mid N=n)=P(Y\in A).$$