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In most real scenarios I know the random variable X is bounded.

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    $\begingroup$ Note that "finite variance" and "bounded" don't mean the same thing. For an example of a real-life unbounded random variable, suppose that a coin is flipped repeatedly and define $X$ to be the number of flips it takes for the first head to appear. I claim that $X$ is a real-life unbounded random variable (although some people might not agree). $\endgroup$
    – mark999
    Commented Jul 5, 2015 at 11:11
  • $\begingroup$ You're right. You can have a variable X that isn't bounded and have finite variance. But, if you have infinite variance you can't be dealing with a variable that is bounded. This is what I meant to say. $\endgroup$ Commented Jul 5, 2015 at 11:22
  • $\begingroup$ @mark999 I thought a bit more about your example. It's a good example. Maybe you can put it as an answer? $\endgroup$ Commented Jul 5, 2015 at 11:30
  • $\begingroup$ Probably not, see stats.stackexchange.com/questions/94402/… $\endgroup$ Commented Apr 8, 2017 at 15:17

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If the data for a variate are directly recorded in some way (not defined as a ratio of two other variables which are the actual ones stored, for example), the recorded variate will pretty much have to have bounded variance, because we don't have a way of recording an infinite number of figures.

Even without that issue, almost all measured variates are actually bounded in some way even when we model them by distributions that are not. For example, a lognormal distribution is sometimes used for sizes of things (rocks or incomes, for example) and that distribution has no upper bound, but no rock on earth could be bigger than the earth and no income can be bigger than the entire economy of the world.

However, variates that are derived from other variates may not be so clearly bounded, at least in the sense that there's a readily identifiable finite upper limit (in particular situations, there generally will be some limit, but it may be impossible to identify other than knowing it must be astronomically large)

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  • $\begingroup$ 'variates that are derived from other variates' I understand that such scenarios may often occur in Economics or Finance. But, wouldn't you agree that for most hard sciences the variates are clearly bounded? I think that most hard sciences develop from the approach defined in your first paragraph. Experiments are done, data is recorded, and from that theories are developed. $\endgroup$ Commented Jul 5, 2015 at 11:06
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    $\begingroup$ (+1) I would disagree with that, Aidan. To echo Glen's answer, it is important to make a distinction between a variable--which is a mathematical construct in a model--and data. The variance of a variable makes sense, but expectations of functions of data do not: the data are just collections of numbers. The most common scientific applications of statistical theory use unbounded variables (which have Normal, Poisson, Gamma, and other unbounded distributions). Thus your claim that data often are bounded looks valid, but it is questionable whether "the variates are clearly bounded." $\endgroup$
    – whuber
    Commented Jul 5, 2015 at 13:29
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    $\begingroup$ Boundedness in principle without some hard, upper bound is virtualy without any value: stats.stackexchange.com/questions/94402/… $\endgroup$ Commented Apr 8, 2017 at 15:18
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The Cauchy distribution https://en.wikipedia.org/wiki/Cauchy_distribution is unbounded and does not have finite moments (mean, variance, etc.) of any positive integer order. Note that the Cauchy Principal Value https://en.wikipedia.org/wiki/Cauchy_principal_value of the mean exists and is equal to the median, and if this is used, the variance = infinity.

Per the Wikipedia article "The Cauchy distribution is the distribution of the X-intercept of a ray issuing from (x_0, gamma) with a uniformly distributed angle. Its importance in physics is the result of it being the solution to the differential equation describing forced resonance. In mathematics, it is closely related to the Poisson kernel, which is the fundamental solution for the Laplace equation in the upper half-plane. In spectroscopy, it is the description of the shape of spectral lines which are subject to homogeneous broadening in which all atoms interact in the same way with the frequency range contained in the line shape."

Per The Existence of the Moments of the Cauchy Distribution under a Simple Transformation of Dividing with a Constant, "Because of this special nature, some authors consider the Cauchy distribution as a pathological case. However, it can be postulated as a model for describing data that arise as $n$ realizations of the ratio of two normal random variables. Other applications given in literature: [4] found that Cauchy distribution describes the distribution of velocity differences induced by different vortex elements. An application of the Cauchy distribution to study the polar and non-polar liquids in porous glasses is given in [5]. In [1] pointed out that the Cauchy distribution describes the distribution of hypocenters on focal spheres of earthquakes. It is shown in the paper [7] that the source of fluctuations in contact window dimensions is variation in contact resistivity, and the contact resistivity is distributed as a Cauchy random variable."

I think that some of these qualify as "real-life" applications. It may be fair to say that most, but not all, real-life distributions have finite variance.

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A short supplementary answer/example of real life distributions with unbounded variance: an exponential random variable $X \sim \text{Exponential}(1)$, the random variable $Y =\frac{1}{X}$ will have unbounded variance.

Tangential note: an exponential distribution is often used to model waiting times, i.e., time per event. The reciprocal thus has a meaningful interpretation, i.e., event per time.

Quoting Wikipedia:

The reciprocal exponential distribution finds use in the analysis of fading wireless communication systems.

RE @whuber's comments for posterity: the original example I used was $X$ is a random variable taking values in $[-1, 1]$, this is insufficient (I meant to say a uniform(-1, 1) random variable).

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    $\begingroup$ The conclusion is incorrect. The simplest counterexample is a Rademacher variable (where $X=1/X$ almost surely). For a more interesting counterexample whose support is the entire interval $[-1,1],$ take a variable with density $f(x)=(3/2)x^2$ on that interval. For a more careful statement, please see stats.stackexchange.com/a/299765/919. $\endgroup$
    – whuber
    Commented Aug 23, 2023 at 14:09
  • $\begingroup$ Thanks @whuber, I changed the example to inverse exponential. $\endgroup$
    – fool
    Commented Aug 23, 2023 at 19:04
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    $\begingroup$ If you originally meant a uniform distribution that includes 0 in its support, then the conclusion was correct, for the reasons given in the post I referenced. But you only indicated that the distribution was "taking values in [-1,1]." // More importantly, your point is a good one. Consider, for instance, any positive random variable $X.$ It's perfectly fine to re-express its values using, say, a Box-Cox transformation. And if $X$ has some chance of being close to $0,$ there will exist a Box-Cox re-expression of $X$ with infinite variance. $\endgroup$
    – whuber
    Commented Aug 23, 2023 at 19:16
  • $\begingroup$ Edited the post again to reflect your comments. I'm gonna have to read the reference you shared later (I should have done my due diligence to understand the reason why it was incorrect). For now, I am inclined to keep this new example, the inverse exponential random variable, because the exponential random variable occurs quite naturally in the real world (e.g., waiting times), and its reciprocal also has an interpretable meaning. $\endgroup$
    – fool
    Commented Aug 23, 2023 at 20:09
  • $\begingroup$ Yes, I didn't mean to suggest the new example was wrong: it's perfectly fine. $\endgroup$
    – whuber
    Commented Aug 23, 2023 at 22:12
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Maybe most, whatever that means ... but it is not safe to assume that random variables in practical problems have finite variance. Note that random variables, variances and so on are theoretical concepts, they belong to models, not to empirical data.

For example, nobody can be richer than the totality of humanitys properties, giving a safe upper bound, but which is obviously too high. If we take that as an absolute upper bound, it would seem that variance must be finite. But, in reality, a lot of wealth is in the upper parts of that distribution, and any hard upper bound is unrealistic. What really matters is the upper tail behavior, and that might be better described by a distribution with infinite variance. For more details see https://stats.stackexchange.com/a/100161/11887

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