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In explaining why uncorrelated does not imply independent, there are several examples that involve a bunch of random variables, but they all seem so abstract: 1 2 3 4.

This answer seems to make sense. My interpretation: A random variable and its square may be uncorrelated (since apparently lack of correlation is something like linear independence) but they are clearly dependent.

I guess an example would be that (standardised?) height and height$^2$ might be uncorrelated but dependent, but I don't see why anyone would want to compare height and height$^2$.

For the purpose of giving intuition to a beginner in elementary probability theory or similar purposes, what are some real-life examples of uncorrelated but dependent random variables?

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    $\begingroup$ This doesn't answer your question, but seems relevant: Sometimes a rv and its square are correlated and sometimes uncorrelated. For example, if X is uniform on [0,1], then X and X^2 are uncorrelated. But if X is uniform on [-1, 1], then X and X^2 are uncorrelated. (Draw a picture to help see this.) However, in both cases, X and X^2 are dependent. $\endgroup$ – Martha Dec 26 '15 at 5:39
  • $\begingroup$ @Martha there's a typo in your comment. I think it's the first 'uncorrelated' that should be 'correlated'. ;) $\endgroup$ – An old man in the sea. Dec 26 '15 at 11:50
  • $\begingroup$ @Anoldmaninthesea correlated and sometimes correlated? $\endgroup$ – BCLC Dec 26 '15 at 12:18
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    $\begingroup$ @BCLC "if X is uniform on [0,1], then X and X^2 are uncorrelated." Should be "if X is uniform on [0,1], then X and X^2 are correlated.", I think. $\endgroup$ – An old man in the sea. Dec 26 '15 at 12:25
  • $\begingroup$ @Anoldmaninthesea You are correct: Correlated on [0,1], but uncorrelated on [-1,1]. Thanks for pointing out the typo. $\endgroup$ – Martha Dec 27 '15 at 20:44
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In finance, GARCH (generalized autoregressive conditional heteroskedasticity) effects are widely cited here: stock returns $r_t:=(P_t-P_{t-1})/P_{t-1}$, with $P_t$ the price at time $t$, themselves are uncorrelated with their own past $r_{t-1}$ if stock markets are efficient (else, you could easily and profitably predict where prices are going), but their squares $r_t^2$ and $r_{t-1}^2$ are not: there is time dependence in the variances, which cluster in time, with periods of high variance in volatile times.

Here is an artificial example (yet again, I know, but "real" stock return series may well look similar):

enter image description here

You see the high volatility cluster around in particular $t\approx400$.

Generated using

library(TSA)
garch01.sim <- garch.sim(alpha=c(.01,.55),beta=0.4,n=500)
plot(garch01.sim, type='l', ylab=expression(r[t]),xlab='t')
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  • $\begingroup$ Thanks valiant pungent reindeer king Hanck. A little rigour please? ^-^ By stock returns do you mean Rt= (St+1-St)/St? Squares of St or squares or Rt? $\endgroup$ – BCLC Dec 26 '15 at 17:20
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    $\begingroup$ I added a little clarification $\endgroup$ – Christoph Hanck Dec 27 '15 at 13:17
  • $\begingroup$ Is that R? $ \ $ $\endgroup$ – BCLC Feb 18 '17 at 12:20
  • $\begingroup$ It is R. It requires the package TSA. $\endgroup$ – toliveira May 22 at 1:03
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A simple example is a bivariate distribution that is uniform on a doughnut-shaped area. The variables are uncorrelated, but clearly dependent - for example, if you know one variable is near its mean, then the other must be distant from its mean.

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  • $\begingroup$ What exactly are the two variables? $\endgroup$ – BCLC Dec 27 '15 at 0:42
  • $\begingroup$ The two random variables $X$ and $Y$ whose joint distribution is uniform on the doughnut. For a specific example, consider the joint density $f(x,y) = 1 / 3\pi$ when $1 < x^2+y^2 < 2$ and $0$ otherwise. $\endgroup$ – rvl Dec 27 '15 at 0:55
  • $\begingroup$ Well I guess physics examples are real life. Thanks rvl. Why is your example true? $\endgroup$ – BCLC Dec 27 '15 at 2:43
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    $\begingroup$ Draw a graph of the region where the density is nonzero and think about it. $\endgroup$ – rvl Dec 27 '15 at 3:18
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I found the following figure from wiki is very useful for intuition. In particular, the bottom row show examples of uncorrelated but dependent distributions.

Caption of the above plot in wiki: Several sets of (x, y) points, with the Pearson correlation coefficient of x and y for each set. Note that the correlation reflects the noisiness and direction of a linear relationship (top row), but not the slope of that relationship (middle), nor many aspects of nonlinear relationships (bottom). N.B.: the figure in the center has a slope of 0 but in that case the correlation coefficient is undefined because the variance of Y is zero.

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