The discussion revolves around the relationship between the expectation of a random variable E[X] and the expectation of its absolute value E[|X|], particularly in the context of continuous random variables. Participants explore definitions and implications of these expectations, especially in cases involving Cauchy distributions.
Participants express differing views on the existence of E[X] for Cauchy distributions and the implications of E[|X|] being infinite. The discussion remains unresolved, with multiple competing interpretations of the definitions and properties of expectation.
There are limitations regarding the definitions of expectation and the conditions under which they apply, particularly in relation to improper integrals and specific distributions like the Cauchy distribution.
Ok, thanks. Looking back in our textbook, the definition of "expectation" begins: "Suppose \int_\Omega |X(\omega)| dP(\omega) <\infty. Then we define the expectation..." So I guess that's the way it is!mathman said:It is a subtle point. If E(|X|) is infinite, then according to some definitions, E(X) does not exist.
On the other hand if the distribution is Cauchy { density 1/[π(1+x2)]}, then E(X)=0 by symmetry even though E(|X|) is infinite.
mathman said:It is a subtle point. If E(|X|) is infinite, then according to some definitions, E(X) does not exist.
On the other hand if the distribution is Cauchy { density 1/[π(1+x2)]}, then E(X)=0 by symmetry even though E(|X|) is infinite.