Let X1,X2,...,Xn be iid with the following pdffx (x|θ)= 1/2θe−|x|/θ for all x∈R,where θ > 0. (a) Compute the moment generating function (m...

Show that the binomial moment generating function converges to the Poisson moment generating function if we let $n \rightarrow \infty$ and $p \rightarrow 0$ in such a way that $n p$ approaches a value $\mu>0 .[$Hint$:$ Use the calculus theorem that was used in showing that the binomial pmf converges to the Poisson pmf.] There is, in fact, a theorem saying that convergence of the mgf implies convergence of the probability distribution. In particular, convergence of the binomial mgf to the Poisson mgf implies $b(x ; n, p) \rightarrow p(x ; \mu) .$