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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 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?
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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
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Nov
4
accepted If $\mathrm{E} |X|^2$ exists, then $\mathrm{E} X$ also exists
Nov
3
answered For $E (X - EX)^2$ to exist, do we need $EX$ to exist and be finite?
Nov
3
comment For $E (X - EX)^2$ to exist, do we need $EX$ to exist and be finite?
@Glen_b: (1) do you mean that the even central moments exist iff $EX$ exists? (2) I understand you use a different definition in your stat.se reply, and I updated that in my question. Now I hope for some help with what I miss when comparing two other different ways.
Nov
3
revised For $E (X - EX)^2$ to exist, do we need $EX$ to exist and be finite?
added 416 characters in body
Nov
3
comment For $E (X - EX)^2$ to exist, do we need $EX$ to exist and be finite?
@Glen_b: (1) Can central moments of other orders, i.e. $E (X - EX)^n, n \in \mathbb N$, be defined similarly? Or is variance the only case (or one of a few cases) that can be defined without refering to $EX$? (2) About references, I haven't seen such definition before, although I admit I don't read much. So I would like to know more, for example, to answer the questions in (1) and many other questions I have and will have.
Nov
3
comment For $E (X - EX)^2$ to exist, do we need $EX$ to exist and be finite?
@Glen_b: Thanks. do you mean $Var X := \frac{1}{2} \int (X_1-X_2)^2 d P$, where $X_1$ and $X_2$ are iid with the same distribution of $X$? That definition does make sense, and always exist (whether it is finite or infinite). Since we have more than one definitions, which definition of variance is used in rigorous probability theory? In what references do you see that definition?
Nov
3
comment If $\mathrm{E} |X|^2$ exists, then $\mathrm{E} X$ also exists
Hi Michael, I think we might have learned from similar references before, about existence of lebesgue integral and finitness of lebesgue integral. I am confused about other peoples posts here stats.stackexchange.com/questions/91512/… and here math.stackexchange.com/questions/1004772/…. I would like to know your thoughts. Thanks.