Mathematical Modeling in Finance with Stochastic Processes

128 CHAPTER 4. LIMIT THEOREMS FOR STOCHASTIC PROCESSES

Sources

This section is adapted from Chapter 8, “Limit Theorems”,A First Course
in Probability, by Sheldon Ross, Macmillan, 1976.

Problems to Work for Understanding

1. SupposeXis a continuous random variable with mean and variance
   both equal to 20. What can be said aboutP[0≤X≤40]?


2. Suppose X is an exponentially distributed random variable with mean
   E[X] = 1. Forx= 0.5, 1, and 2, compareP[X≥x] with the Markov
   inequality bound.


3. Suppose X is a Bernoulli random variable withP[X= 1] = pand
   P[X= 0] = 1−p=q. CompareP[X≥1] with the Markov inequality
   bound.


4. Let X 1 ,X 2 ,...,X 10 be independent Poisson random variables with
   mean 1. First use the Markov Inequality to get a bound onP[X 1 +···+X 10 >15].
   Next find the exact probability thatP[X 1 +···+X 10 >15] using that
   the fact that the sum of independent Poisson random variables with
   parametersλ 1 ,λ 2 is again Poisson with parameterλ 1 +λ 2.

Outside Readings and Links:

1. Virtual Laboratories in Probability and Statistics. Search the page
   for Weak Law and then run the Binomial Coin Experiment and the
   Matching Experiment.

2.

4.2 Moment Generating Functions

Rating

Mathematically Mature: may contain mathematics beyond calculus with
proofs.