Find conditions on $\lambda, n, p$, so that the characteristic function of the Binomial converges to that of the Poisson
Binomial distribution is given as $P(X=k)=\binom{n}{k}p^k(1-p)^{n-k}$
Poisson distribution is given as $P(X=k)=\frac{\lambda^ke^{-\lambda}}{k!}$
So characteristic function of binomial: $\phi_X(t)=E(e^{itX})=\sum_1^n \binom{n}{k} e^{itk} p^k(1-p)^{n-k}=\sum \binom{n}{k} (e^{it}p)^k (1-p)^{n-k}=(pe^{it}+(1-p))^n$
and that of Poisson in a similar way:$\quad$$e^{\lambda(e^{it}-1)}$
Now my questions:
1) We defined Expectation with integral, why here with a sum ?
2) Can you give me a hint to solve the problem ?
Best Answer
Concerning your questions