Understanding why a sequence of random variables is not identically distributed Consider the sequence $(X_n)_n$ of mutually independent v.a. whose probability law is defined for
$P(X_n=-\sqrt n)= P(X_n=\sqrt n)= \frac{1}{2}$
Why they are not identically distributed?
 A: Hint: Is $\mathbb{P}(X_n = \sqrt{n}) = \mathbb{P}(X_m = \sqrt{n})$ for $n\neq m$?
Recall that two random variables $X$, $Y$ are said to be identically distributed if for every Borel set $B$ we have that $\mathbb{P}(X\in B)=\mathbb{P}(Y\in B)$, or if we define the distribution associated to a random variable $Z$ as $\mathbb{P}_Z:\beta(\mathbb{R})\to [0,1]$ given by $\mathbb{P}_Z(B) = \mathbb{P}(Z\in B)$ then we can express the previous condition as $\mathbb{P}_X=\mathbb{P}_Y$.
This can be cumbersome to prove, as there are many Borel sets. However, it is a well-known fact that the distribution functions associated to two random variables coincide if and only if their cummulative distribution functions coincide, where for a random variable $Z$ its cummulative distribution function is $F_Z:\mathbb{R}\to[0,1]$ given by $F_Z(t)=\mathbb{P}_Z((-\infty, t])$.
Proving that the cummulative distributions coincide or do not coincide is a simple way of checking if two random variables are identically dostributed, but of course in some cases such as this one one need not do all of those calculations to convince oneself of the fact because the distributions do not agree in some simple Borel sets.
Note that $$F_{X_n}(t)=\begin{cases}0 &\text{ if } t< -\sqrt{n}\\ \frac{1}{2} &\text{ if }-\sqrt{n} \leq t< \sqrt{n}\\ 1 &\text{ if } \sqrt{n}\leq t \end{cases}$$
which gives different functions for every $n$ thus confirming (and proving) the assertion.
A: For discrete random variables (as the ones you have given) to be identically distributed, they must have the SAME PROBABILITY of hitting the SAME OUTCOMES.
In your case, you might be tricked into thinking that they are identically distributed because $P=\frac{1}{2}$ always, but notice that the variables actually CAN'T TAKE THE SAME VALUES. For example, $X_1$ can only be $+1$ or $-1$, and nothing else, whereas $X_4$ can only be $+2$ or $-2$.
Identical distribution doesn't just mean that they all have two equally likely outcomes; the outcomes themselves matter. A 50/50 game of Russian roulette is different than a 50/50 fair coin toss.
