# When is a process IID?

I am unable to solve this homework question and was hoping to get some help for part (a).

I know that $Z$ is a stationary process because its generating function does not change with time. By the opposite reasoning, $W$ is not stationary because the coin flip happens at every time $n$.

For the second part of (a), I had thought that $Z$ would be IID because the probability of the events are independent and occur with the same distribution. $W$ is not IID because of the coin flip occurring at every time $n$ which means its distribution at time $n$ is different than all other times.

The answer turned out to be the opposite but I don't understand why.

PS: Don't worry its not an actual homework https://documents.epfl.ch/groups/i/ip/ipg/www/2015-2016/Information%20Theory%20and%20Coding/hw5.pdf

The process $W$ is indeed i.i.d., and $Z$ is not, unless the distribution of $\{X_n\}_{n=-\infty}^\infty$ is the same as the distribution of $\{Y_n\}_{n=-\infty}^\infty$.
Let us assume the distribution of $\{X_n\}_{n=-\infty}^\infty$ is not the same as the distribution of $\{Y_n\}_{n=-\infty}^\infty,$ so there is some set $A$ of values of those random variables for which $\Pr(X_n\in A)\ne\Pr(Y_n\in A).$ No generality is lost by assuming $\Pr(X_n\in A)>\Pr(Y_n\in A).$ Now suppose you've observed $Z_n$ for $100$ values of $n,$ and in every case you saw $Z_n\in A.$ What is the conditional probability that $Z_m\in A,$ where $m$ is some index not among those $100$ for which $Z_n$ was observed? The answer is that it is greater than the unconditional, or "marginal" probability that $Z_m\in A,$ because the observation all of those outcomes were in $A$ makes it more probable that $\Theta=1$ than it would otherwise have been. The terms in the sequence $\{Z_n\}$ are _conditionally independent given that $\Theta=0,$ or that $\Theta=1,$ but unconditionally they are not independent.
On the other hand the sequence $W$ is i.i.d. because for every value of $n$ you choose $\Theta_n,$ $X_n,$ and $Y_n,$ and consequently $W_n,$ independently of the results for all other values of $n,$ with the same probability distributions. Specifically $$\Pr(W_n\in A) = \Pr(W_n\in A\mid\Theta=1)\Pr(\Theta=1) + \Pr(W_n\in A\mid\Theta=0)\Pr(\Theta=0)$$ and the event $W_n\in A$ has that same probability for each value of $n$ separately.