Expected Value of a Random Variable

Statement

Definition. (Ost20, Definition 6.1.1) Let $(\Omega ,\mathcal{F},P)$ denote a probability space, $(\mathcal{X},\mathcal{S})$ be a measurable space, and let $X:\Omega \to \mathcal{X}$ denote a random variable. The expectation (or expected value) of $X$ is defined as

$$E(X):=\sum _{x\in \mathcal{X}}x{P}_X(x),$$

when $X$ is a discrete random variable with PMF $P_X$ and as

$$E(X):={\int }_{x\in \mathcal{X}}x{P}_X(x)dx,$$

when $X$ is a continuous random variable with PDF $P_X$.

The expectation of a random variable is said to exist, if it is finite

Remark. (Ost20, Section 6.3) The theoretical constructs of expectations, variances, and standard deviations should not be confused with the concepts of sample means, sample variance, and sample standard deviations. The former entities are of theoretical nature and can be evaluated once the distributions of random variables have been specified. The latter entities are of practical nature, can be evaluated numerically based on observed data, and serve as estimators of the former theoretical quantities

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Definition. (Ost20, Definition 6.3.1) Let $(X_1,\dots ,X_n)$ be random variables. Then the sample mean of $(X_1,\dots ,X_n)$ is defined as the arithmetic average

$$\mu =\frac{1}{n}\sum _{i=1}^n{X}_i.$$