Let
$$ x_k = \begin{cases} 1 & \mathrm{prob} (1/2)\\ -1 &\mathrm{prob} (1/2) \end{cases}$$
be independient random variables.
Show that the characteristic function of the random variable
$$ \frac{x_1 + x_2 + x_3 + ... + x_n}{\sqrt{n}}$$ is equal to $\left(\cos\frac{w}{\sqrt{n}}\right)^n $
The characteristic function is defined as
$$ \phi_x(w) = \sum e^i{iwx}f(x)$$
for discrete variables.
Then we have
$$ E(e^{iwX}) = e^{iw(1)}\left(\frac{1}{2}\right) + e^{iw(-1)}\left(\frac{1}{2}\right) = \frac{1}{2}(e^{iw}+e^{-iw}) = \cos w .$$
And that's where I'm stuck. Thanks for your help.
If $Y_j,1\leqslant j\leqslant n$ are independent random variable with respective characteristic functions $\varphi_j$, then the characteristic function of $\sum_{j=1}^nY_j$ is given by $\prod_{j=1}^n\varphi_j(w)$.
Applying this to $Y_j:= n^{-1/2} x_j$, we obtain that the characteristic function $\phi_n$ of $n^{-1/2}\sum_{j=1}^nx_j$ is equal to $$\phi_n(w)=\prod_{j=1}^n\mathbb E\left[e^{iwn^{-1/2} x_j}\right].$$ Since the random variables $x_j$ are identically distributed, this reduces to $\phi_n(w)=\left(\mathbb E\left[e^{iwn^{-1/2} x_1}\right]\right)^n$.