Consider the i.i.d (independent identically distributed) sequence $X_1,X_2,X_3,..$ of random variables such that $X_i \in {1,2,3,...}$ and for all $i$ $P(X_n=i) = p_i > 0$
Let $Y_n = 1$ with probability 1. For $n >= 2 $ let $Y_n = 1$ if the value of $X_n$ has not been observed previously; and $Y_n=0$ otherwise.
Are variables $Y_1, Y_2, Y_3,..$ independent? Are they identically distributed ?
The answer to both of these questions is apparently no (not independent nor identically distributed), and I'm wondering why. This can be shown via a counter-example (all you need is one), but I'm still unclear about this.
I would appreciate if somebody from the community could clearly explain why the Ys are not independent nor identically distributed.