4: Expected Value

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Expected value is one of the fundamental concepts in probability, in a sense more general than probability itself. The expected value of a real-valued random variable gives a measure of the center of the distribution of the variable. More importantly, by taking the expected value of various functions of a general random variable, we can measure many interesting features of its distribution, including spread, skewness, kurtosis, and correlation. Generating functions are certain types of expected value that completely determine the distribution of the variable. Conditional expected value, which incorporates known information in the computation, is one of the fundamental concepts in probability.

In the advanced topics, we define expected value as an integral with respect to the underlying probability measure. We also revisit conditional expected value from a measure-theoretic point of view. We study vector spaces of random variables with certain expected values as the norms of the spaces, which in turn leads to modes of convergence for random variables.

4.1: Definitions and Basic Properties
4.2: Additional Properties
4.3: Variance
Recall the expected value of a real-valued random variable is the mean of the variable, and is a measure of the center of the distribution. Recall also that by taking the expected value of various transformations of the variable, we can measure other interesting characteristics of the distribution. In this section, we will study expected values that measure the spread of the distribution about the mean.
4.4: Skewness and Kurtosis
4.5: Covariance and Correlation
4.6: Generating Functions
4.7: Conditional Expected Value
4.8: Expected Value and Covariance Matrices
4.9: Expected Value as an Integral
4.10: Conditional Expected Value Revisited
4.11: Vector Spaces of Random Variables
4.12: Uniformly Integrable Variables
4.13: Kernels and Operators