I How does E[X] and E[|X|] relate?

[MidgetDwarf]

TL;DR

For expectations of random variables. Relationship between E[X] n E[|X|]?

Greetings,

I am studying probability theory [non-measure theory] from a textbook.

I stumbled to the topic stating that Cauchy Distribution has no moments.

It was not proved, and I tried working it via direct calculation of the improper integral of E[X^n] for the case n=1.

Anyhow, I wanted to generalize this without success. I stumbled upon this thread here:

I Thread 'How to prove the Cauchy distribution has no moments?'

Aug 14, 2020

How can I prove the Cauchy distribution has no moments?

$E(X^n)=\int_{-\infty}^\infty\frac{x^n}{\pi(1+x^2)}\ dx.$

I can prove myself, letting $n=1$ or $n=2$ that it does not have any moment. However, how would I prove for ALL $n$, that the Cauchy distribution has no moments?

• Neothilic
• Cauchy Distribution Expectation Moments
• Replies: 5
• Forum: Set Theory, Logic, Probability, Statistics

I really enjoyed the proof given StoneTemplePython. However, I am unsure of the following:

Why does E[|x|] diverges implies E[x] diverges?I believe this is the crux of the poof. Since it assumes, in my understanding, that having shown E[|X|] diverges, then applying the comparison test for improper integrals in the last line of that thread, then E[|X|^n] diverges.

How does this relate to E[X^n]?

. My hunch tells me that it has something to do with f is integrable iff |f| is integrable, and how we can decompose |f| into the positive and negative parts of f?

My knowledge of Measure Theory is very poor. So hopefully someone here can why.

 

[anuttarasammyak]

Integral for even n diverges clearly to plus infinite. Integral for odd n is zero for integrand [-L,L]. However we do not have such a constraint thus the integral diverges plus or minus.

 

[FactChecker]

Let $X_-$ and $X_+$ denote the negative and positive parts of $X$. Then $E[X]=E[X_+]+E[X_-]$ and $E[|X|]=E[X_+]-E[X_-]$. If $E[|X|]$ is infinite or undefined, then one or both of $E[X_+]$ and $E[X_-]$ are infinite, so $E[X]=E[X_+]+E[X_-]$ also diverges.
I'm not sure about the higher orders, $E[X^n]$ for $n\gt 1$. If $X$ is restricted to $|X| \lt 1$, then those moments may exist.