On the New Properties of Conditional Expectations

Accepted - May 2024

Abstract

The concept of conditional expectation is important in applications of probability and statistics in many areas such as reliability engineering, economy, finance, and actuarial sciences due to its property of being the best predictor of a random variable as a function of another random variable. This concept also is essential in the martingale theory and theory of Markov processes. Even though, there has been studied and published many interesting properties of conditional expectations with respect to a sigma- algebra generated by a random variable it remains an attractive subject having interesting applications in many fields. In this paper, we present some new properties of the conditional expectation of a random variable given another random variable and describe useful applications in problems of per-share price of stock markets. The copula and dependence properties of conditional expectations as random variables are also studied. We present also some new equalities having interesting applications and results in martingale theory and Markov processes.