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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...

Question

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 (mgf) for thisdistribution.(b) Find the first moment, E[X].(c) Find the second moment, E[X2].

Let X1,X2,...,Xn be iid with the following pdf fx (x|θ)= 1/2θe−|x|/θ for all x∈R, where θ > 0. (a) Compute the moment generating function (mgf) for this distribution. (b) Find the first moment, E[X]. (c) Find the second moment, E[X2].



Answers

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 $b(x ; n, p) \rightarrow p(x ; \mu) .$


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