Why does iid (independent and identically distributed) have same probability distribution. What does it implies.
Let us consider a simple example: imagine we flip a coin several times. Let the random variable $Y_n = 1$ if trial $n$ is heads, $Y_n=0$ otherwise.
$Y_i$ and $Y_j$ are independent as the result of trial $j$ does not depend at all of the result of trial $i$.
As we repeat same experiment, we say that $Y_i$ identically distributed. We have same probability to observe heads $ P(Y_i=1)=P(Y_j=1)$ (and obviously tails $P(Y_i=0)=P(Y_j=0))$.