Limits of Moment-Generating Functions

It is possible to approximate binomial probabilities using the Poisson distribution. This is proved by showing that the limit of the binomial moment-generating function converges to the Poisson moment-generating function. A proof of the Central Limit Theorem involves the limit of moment-generating functions converging to the N(0, 1) moment-generating function.

1. Limits of binomial moment-generating functions and comparisons of binomial and Poisson probability histograms. Solution

2. Let [Maple Math] , ..., [Maple Math] be a random sample of size n from an exponential distribution with mean [Maple Math] . Show that moment-generating function of [Maple Math] converges to the moment-generating function for the N (0, 1) distribution. Solution

3. Let [Maple Math] , ..., [Maple Math] be a random sample of size n from a U -shaped distribution with p.d.f. [Maple Math] , [Maple Math] < x < 1. Show that the moment-generating function of [Maple Math] = [Maple Math] converges to the moment-generating function for the N(0,1) distribution. Solution