Mathematical Foundations 2

This chapter discusses basic mathematical concepts, probability, random variables, and convergence of random variables, used in stochastic differential equations. It defines a random experiment as a class of occurrences that can happen repeatedly, for an unlimited number of times, under essentially unchanged conditions. A moment of a random variable X is defined as the expected value of some particular function of X. Specifically, moments of a random variable X are specified in terms of having either zero or E(X) as the reference point. For X and Y discrete random variables, the summation of the bivariate probability mass function f (X, Y) over all Y within Yj, j = 1,…, m yields the univariate probability mass function g(X) called the marginal probability mass function of X. The conditional probability density function of X given Y = y is obtained from the joint and marginal densities.