134 CHAPTER 4. LIMIT THEOREMS FOR STOCHASTIC PROCESSES
An alternative visual proof that the sum of independent normal random
variables is again a normal random variable using only calculus is The Sum
of Independent Normal Random Variables is Normal
Sources
This section is adapted from:Introduction to Probability Models, by Sheldon
Ross.
Problems to Work for Understanding
- Calculate the moment generating function of a random variable X
having a uniform distribution on [0,1]. Use this to obtainE[X] and
Var [X]. - Calculate the moment generating function of a discrete random vari-
ableXhaving a geometric distribution. Use this to obtainE[X] and
Var [X].
Outside Readings and Links:
- http://www.math.uah.edu/stat/expect/Generating.xhtml
- http://mathworld.wolfram.com/Moment-GeneratingFunction.html in Math-
World.com
4.3 The Central Limit Theorem
Rating
Mathematically Mature: may contain mathematics beyond calculus with
proofs.
Section Starter Question
What is the most important probability distribution? Why do you choose
that distribution as most important?