Random variable from characteristic function

Question

I need to find the random variable knowing its characteristic function. I've found various post related to the problem, but the matter is that I cannot manipulate complex integral, neither I've studied Fourier transforms.

So I ask if there exist an easier way to understand to which r.v. belong this c.f.:

$$\phi(t)=\frac{1}{16}(e^{3it}+1)+\frac{1}{4}(e^{it}+e^{\frac{2}{5}it})+\frac{3}{8}e^{7it}.$$

Thanks for your help.

KB

Answer 1

This is a weighted (with positive weights and sum $1$) sum of functions of the form $t\mapsto e^{iat}$, where $a$ is a real number. If $\varphi_X(t)=\sum_{j=1}^np_je^{ia_jt}$ where $p_j\geqslant 0$ and $\sum_{j=1}^np_j=1$, then $\mathbb P(X=a_j)=p_j$ for each $j\in \{1,\dots,n\}$ (the converse is also true).