What is law of random variable $S_n = X_1 + \cdots + X_n$

We consider a sequence $$(X_n)_{n \leq 1}$$ of mutually independent random variables defined on a probability space $$(\Omega, \mathcal{A}, \mathbb{P})$$ with value in $$\{-1,1\}$$ and such that, for all $$k \geq 1$$ $$\mathbb{P} (X_k = -1) = \mathbb{P} (X_k = 1) = \frac{1}{2}$$ For all $$n \geq 1$$, we set $$S_n = X_1 + \cdots + X_n$$

Problem

What is law of random variable $$S_n$$?

• $X= 2Y-1$ where $Y$ has the distribution $P(Y=0)=P(Y=1)=\frac{1}{2}.$ The distribution of the sum of $Y$'s is well-known. Commented Nov 3, 2020 at 22:40
Ignoring order, all drawings with $$p$$ positives are equiprobable with probability $$\dfrac1{2^p}\dfrac1{2^{n-p}}=\dfrac1{2^n}$$, and the values sum to $$p-(n-p)=2p-n$$. Observe that all sums are distinct, hence a function of $$p$$.
Now if we restore order, the $$p$$ positives can appear in $$\displaystyle\binom np$$ different ways, and this gives us the distribution:
$$\mathbb P(S_n=2p-n)=\binom np\frac1{2^n}$$ for $$p=0,1,\cdots n$$.
Hint: The Count for values of $$+1$$ among the sequence of $${(X_k)}_{k=1}^n$$ shall be the count for successes among $$n$$ independent Bernoulli trials with an identical success rate.