Computer Simulation

At times in this course results of probability theory will be investigated by computer simulation. That is, we create a large quantity of random numbers from the distributions of interest and plot or perform calculations with them to check that they are similar to our probability calculations. This suggests two important questions:

• •

  Q: Given a random distribution, how do we create random numbers from it?

• •

  A: The mathematical theory will be developed in Chapters 4 and 8. To generate random numbers in practice we use a computer package such as R. Appendix A describes the R commands to do this.

• •

  Q: Do a large number of random numbers have similar properties (e.g. mean value) to probability calculations (e.g. expectation)?

• •

  A: Yes (given some assumptions). This is due to a central theorem of probability, the law of large numbers, which is proved (partly) in Chapter 9.

One portion of Math245 will exploit the law of large numbers to approximate intractable integrals and expectations as accurately as desired.