Probability, Random Variables, and Random Processes: Theory and Signal Processing Applications

D

Integration and Integrals

In this appendix, we provide a description of different types of integration for computing expectations of random variables. The Riemann integral is briefly reviewed, so that the more general (and less familiar) Riemann–Stieltjes integral can be described. We will not attempt to be as rigorous as most textbooks on measure theory and integration. Instead, for the probability space , we would like to provide a basic understanding of the meaning of the following integral:

(D.1)

where FX(x) is the cumulative distribution function (cdf) of X, and g(x) is the function to be integrated. We also briefly describe the Lebesgue integral, which is more general than the Riemann integral and can be used for functions that are not Riemann integrable. For the abstract probability space , it is generally written as

(D.2)

where are elements of the sample space.

This material on integration is not necessary for understanding the definition of expectation and the various moments ...