Discrete Distribution Examples

There are times when an infinite series must be evaluated. Some of these problems are easy to solve and others are more challenging. Sometimes it is better to do a problem "by hand" rather than use the computer.

1. Binomial Distribution -- Mean, Variance, Probability Histograms, Normal Approximation

Solution

2. Let X have a Poisson Distribution with mean [Maple Math] .

(a) Find the mean, variance, and other moments of X symbolically.

(b) Both X - bar and the sample variance , [Maple Math] , are unbiased estimators for [Maple Math] . Compare the variances of these two estimators both theoretically and empirically. Solution

3. The power of the probability-generating function (factorial moment-generating function) for determining a p.d.f. can be illustrated with a very simple problem. Let X equal the number of times that an n-sided die must be rolled to observe each face at least once. The probability-generating function of X , as a function of t , is [Maple Math] . A CAS can expand h = h (t) and list the respective probabilities, P ( X = x ), the coefficients of [Maple Math] i n the expansion. Solution