ABSTRACT
As we have seen, for a discrete random variable X with set of possible values A and probability mass function p, E(X) is defined by ∑ x ∈ A x p ( x ) $ \sum _{x\in A}xp(x) $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429457951/63b50632-c513-448c-ba5f-4cb005c69347/content/inline-math10_1.tif"/> . For a continuous random variable X with probability density function f, the same quantity, E(X), is defined by ∫ - ∞ ∞ x f ( x ) d x . $ \int _{-\infty }^{\infty } xf(x) dx. $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429457951/63b50632-c513-448c-ba5f-4cb005c69347/content/inline-math10_2.tif"/> Recall that for a random variable X, the expected value, E(X), might not exist (see Examples 4.18 and 4.19 and Exercise 13, Section 6.3). In the following discussion we always assume that the expected value of a random variable exists.
