91Ó°ÊÓ

Let \({X_n}\)be a random variable having a binomial distribution with parameters n and \({p_n}\)

The m.g.f. of the binomial distribution with parameters n and\({p_n}\)is,

\({\psi _n}\left( t \right) = {\left( {{p_n}\exp \left( t \right) + 1 - {p_n}} \right)^n}\)

If,\(n{p_n} \to \lambda \)

\(\mathop {\lim }\limits_{n \to \infty } \,{\psi _n}\left( t \right) = \mathop {\lim }\limits_{n \to \infty } \,\,{\left( {1 + \frac{{n{p_n}}}{n}\left[ {\exp \left( t \right) - 1} \right]} \right)^n}\)

This converges to\(\exp \left( {\lambda \left[ {{e^t} - 1} \right]} \right)\), which is the m.g.f of the Poisson distribution with mean\(\lambda \)

Hence, the m.g.f. of \({X_n}\) converges to the m.g.f. of the Poisson distribution with mean \(\lambda \)……[Proved]