Hoping someone could show how to use the Characteristic Function of a binomial r.v. to recover its distribution.
Using the inversion formula to recover the pdf of a r.v. with a continuous distribution makes sense to me... but I think I'm just getting thrown off by the discrete cases. Any example would be great.
By definition, the characteristic function of a discrete random variable $X$ with mass distribution $f_X(x)$ over support $\mathcal X$ is: $$\varphi(t) =\mathsf E(\mathsf e^{itX})= \sum_{x\in\mathcal X} f_X(x) e^{itx}$$
The characteristic function of a binomial distribution is: $$\begin{align} \varphi(t) & = ((1-\color{red}{p})+\color{red}{p}\mathsf e^{it})^\color{blue}{n} \\[1ex] & = \sum_{\color{green}{x}=0}^\color{blue}{n} \binom{\color{blue}{n}}{\color{green}{x}}\color{red}{p}^\color{green}{x}(1-\color{red}{p})^{\color{blue}{n}-\color{green}{x}}\mathsf e^{it\color{green}{x}} \end{align}$$
The probability mass distribution is; $$f_X(\color{green}{x}) = \binom{\color{blue}{n}}{\color{green}{x}}\color{red}{p}^x(1-\color{red}{p})^{\color{blue}{n}-\color{green}{x}}$$
I'm ... not seeing any difficulty here.
It's just a matter of expanding the characteristic function into a polynomial series in terms of power $\color{green}{x}$ exponents of $\mathsf e^{it}$