- #1
bpet
- 532
- 7
I'm interested in learning the calculus of general random variables, i.e. those that do not necessarily have a density or mass function - such as mixtures of continuous / discrete / Cantor-type variables.
There seem to be several different approaches:
1. Via densities, using delta functions etc, e.g. [tex]E[X]=\int_{-\infty}^{\infty}x f(x)dx[/tex]
2. Via cumulative distributions, using Stieltjes-type integrals, e.g. [tex]E[X]=\int_{-\infty}^{\infty}xdF(x)[/tex]
3. Via probability measures, e.g. [tex]E[X]=\int x d\mu(x)[/tex]
Each seems to have a well developed rigorous theory. What would be the best approach to focus on, and what's a good accessible book on the subject?
There seem to be several different approaches:
1. Via densities, using delta functions etc, e.g. [tex]E[X]=\int_{-\infty}^{\infty}x f(x)dx[/tex]
2. Via cumulative distributions, using Stieltjes-type integrals, e.g. [tex]E[X]=\int_{-\infty}^{\infty}xdF(x)[/tex]
3. Via probability measures, e.g. [tex]E[X]=\int x d\mu(x)[/tex]
Each seems to have a well developed rigorous theory. What would be the best approach to focus on, and what's a good accessible book on the subject?