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

3. Let X[1], X[2] , ..., X[n] be a random sample of size n from a U -shaped distribution with p.d.f. f(x) = 3/2*x^2 , -1 < x < 1. Show that the moment-generating function of

(Sigma*X[i]-n*mu)/(sigma*sqrt(n)) = (Sigma*X[i]-0)/sqrt(n*3/5)

converges to the moment-generating function for the N(0,1) distribution. Solution