Consider X∼Binomial(n,p)X \sim \text{Binomial}(n, p)X∼Binomial(n,p). What is the characteristic function ϕX(t)\phi_X(t)ϕX(t)?
(peit+1−p)n(pe^{it} + 1 - p)^n(peit+1−p)n
(p+(1−p)eit)n(p + (1-p)e^{it})^n(p+(1−p)eit)n
npeitnpe^{it}npeit
(pet+1−p)n(pe^t + 1 - p)^n(pet+1−p)n