Communication, Networking, Signal and Image Processing (CS)

Question 1: Probability and Random Processes

August 2007

2. (25 Points)

Let $\left\{ \mathbf{X}_{n}\right\} _{n\geq1}$ be a sequence of binomially distributed random variables, with the $n$ -th random variable $\mathbf{X}_{n}$ having pmf $p_{\mathbf{X}_{n}}\left(k\right)=P\left(\left\{ \mathbf{X}_{n}=k\right\} \right)=\left(\begin{array}{c} n\\ k \end{array}\right)p_{n}^{k}\left(1-p_{n}\right)^{n-k}\;,\qquad k=0,\cdots,n,\quad p_{n}\in\left(0,1\right).$

Show that, if the $p_{n}$ have the property that $np_{n}\rightarrow\lambda$ as $n\rightarrow\infty$ , where $\lambda$ is a positive constant, then the sequence $\left\{ \mathbf{X}_{n}\right\} _{n\geq1}$ converges in distribution to a Poisson random variable $\mathbf{X}$ with mean $\lambda$ .

Hint:

You may find the following fact useful:

$\lim_{n\rightarrow\infty}\left(1+\frac{x}{n}\right)^{n}=e^{x}.$