0
$\begingroup$

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

$\endgroup$
3
  • $\begingroup$ 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
1
$\begingroup$

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.

$\endgroup$
0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.