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Just a layman and slow learner. Thank you for your enlightenment and patience, and for giving me the best memory in my life.


Nov
4
comment Independence and uncorrelatedness between two normal random vectors.
Yes. it is defined that way. @MichaelHardy
Nov
4
comment Independence and uncorrelatedness between two normal random vectors.
@MichaelHardy: It is what I think and guess out of my old notes, not an exercise. So I am not claiming my phrasing is correct. Yes, the emphasis is from 1 to 2.
Nov
4
asked Independence and uncorrelatedness between two normal random vectors.
Nov
4
comment Does a Pareto distribution always have variance?
As you mentioned in your comment to my linked reply, the Wikipedia article in question uses "the one that makes a random variable with infinite expectation have undefined variance". But I don't think it is the most standard one, and I am not alone.
Nov
4
asked Are two random vectors independent, iff every pair of components from each vector are independent?
Nov
4
comment Does a Pareto distribution always have variance?
Kevin, Thanks. I realized something, something why Wikipedia wrote about the Parento distrn that way, although I still think you claim that the two are the same is wrong.
Nov
4
comment Does a Pareto distribution always have variance?
I pointed out that they are not, in my last comment and here math.stackexchange.com/questions/1004772/…. I am happy if you can convince me otherwise.
Nov
4
comment For $E (X - EX)^2$ to exist, do we need $EX$ to exist and be finite?
Care to explain what I said wrong?
Nov
4
comment Does a Pareto distribution always have variance?
This is the only post in which I asked why Wikipedia says the variance is infinite for some alpha and does not exist for other alpha. I guess it uses the central moment definition of variance, by which the variance should have existed always. (In the other posts, I either did not ask about specific distribution, or asked about Glen_b's different definition of variance for the specific distribution among other distributions.) Wikipedia doesn't use the definition that Glen_b used, and doesnt agree with the central moment definition either. That is my question here: is Wiki wrong? Or I miss sth?
Nov
4
comment Does a Pareto distribution always have variance?
Not really. Bonjour, so glad to see you again. @Did
Nov
4
comment Does a Pareto distribution always have variance?
well, according to your post, its definition is $∫(x−μ)^2f$, not $∫x^2f−(∫xf)^2$. $ ∞−∞$ doesn't appear in the formmer.
Nov
4
comment Does a Pareto distribution always have variance?
when μ is infinite, the integral in the definition of the variance is also infinite.
Nov
4
revised Does a Pareto distribution always have variance?
added 122 characters in body
Nov
4
comment Does a Pareto distribution always have variance?
For expectation, ∞ does count as "well-defined". If you also look at the Wikipedia link, it says the variance is infty when α∈(1,2], and doesn't exist when α≤1.
Nov
4
asked Does a Pareto distribution always have variance?
Nov
4
comment $l^p$ norm on an inner product space
Thanks. When the inner product space $X$ is infinite-dim, can we still define the "L^p norms" on $X$ as the $L^p$ norms on the coordinate space under a orthonormal basis? Will they be still norms on $X$?
Nov
4
comment For $E (X - EX)^2$ to exist, do we need $EX$ to exist and be finite?
@Glen_b: ok. Then is it right that even order central moments $E(X−EX)^n,n∈N$ exist iff EX exists?
Nov
4
accepted Does $\mathrm{E} |X|$ finite imply $\mathrm{E} |X|^2$ finite?
Nov
4
revised If $\mathrm{E} |X|^2$ exists, then $\mathrm{E} X$ also exists
added 71 characters in body
Nov
4
accepted If $\mathrm{E} |X|^2$ exists, then $\mathrm{E} X$ also exists