When reading about the central limit theorem, the concept of infinite population mean arises.How can a population mean be infinite?

  • $\begingroup$ A population mean can be infinite quite easily, although it does require whatever value you're computing the mean of to be unbounded, obviously. What's the context here? $\endgroup$
    – Brian Tung
    May 20 '15 at 2:13
  • $\begingroup$ The central limit theorem is about the behaviour of the mean of $n$ independent random variables as $n$ increases without limit. But $n$ remains finite. For example, you could imagine throwing a fair die an arbitrarily large number of times and considering the distribution of the average of the throws and how close it is likely to be to $3.5$. $\endgroup$
    – Henry
    May 20 '15 at 6:36
  • $\begingroup$ You should read "infinite-population mean", not "infinite population-mean" ! $\endgroup$
    – user65203
    May 20 '15 at 16:43

Two examples of random variables $Y$ with 'infinite' $E(Y).$

1) Let $X$ be geometric with $P(X = i) = 1/2^i,$ for $ i = 1, 2, \dots.$ Then let $Y = 2^X.$ If you try to find $E(Y)$ you get an infinite sum with all terms equal to 1.

2) Let $X \sim T(1),$ Student's t distribution with one degree of freedom. Then let $Y = |X|.$

Very roughly speaking, the tails need to be 'fat' enough that $\sum |x|f(x)$ or $\int |x|f(x)\,dx$ (taken over all values) is infinite, but not so fat as to prevent $\sum f(x) = 1$ or $\int f(x)\,dx = 1$ (or you wouldn't have a legitimate distribution).

Addendum: Also, relevant to conditions for the CLT: It is possible to have a random variable $Y$ for which $E(Y)$ exists and $Var(Y)$ does not; $T(2)$ is one example.


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