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I read from my textbook that $\text{cov}(X,Y)=0$ does not guarantee X and Y are independent. But if they are independent, their covariance must be 0. I could not think of any proper example yet; could someone provide one?

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You might also enjoy a quick review of Anscombe's Quartet, which illustrates some of the many different ways in which a particular nonzero covariance can be realized by a bivariate dataset. – whuber♦ Jul 11 '11 at 18:01
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The thing to note is that the measure of covariance is a measure of linearity.. Calculating the covariance is answering the question 'Do the data form a straight line pattern?' If the data do follow a linear pattern, they are therefore dependent. BUT, this is only one way in which the data can be dependent. Its like asking 'Am I driving recklessly?' One question might be 'Are you travelling 25 mph over the speed limit?' But that isn't the only way to drive recklessly. Another question could be 'Are you drunk?' etc.. There is more than one way to drive recklessly. – Adam Feb 16 '12 at 4:13

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up vote 15 down vote accepted

Easy example: Let $X$ be a random variable that is $-1$ or $+1$ with probability 0.5. Then let $Y$ be a random variable such that $Y=0$ if $X=-1$, and $Y$ is randomly $-1$ or $+1$ with probability 0.5 if $X=1$.

Clearly $X$ and $Y$ are highly dependent (since knowing $Y$ allows me to perfectly know $X$), but their covariance is zero: They both have zero mean, and

$$\eqalign{ \mathbb{E}[XY] &=&(-1) &\cdot &0 &\cdot &P(X=-1) \\ &+& 1 &\cdot &1 &\cdot &P(X=1,Y=1) \\ &+& 1 &\cdot &(-1)&\cdot &P(X=1,Y=-1) \\ &=&0. }$$

Or more generally, take any distribution $P(X)$ and any $P(Y|X)$ such that $P(Y=a|X) = P(Y=-a|X)$ for all $X$ (i.e., a joint distribution that is symmetric around the $x$ axis), and you will always have zero covariance. But you will have non-independence whenever $P(Y|X) \neq P(Y)$; i.e., the conditionals are not all equal to the marginal. Or ditto for symmetry around the $y$ axis.

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up vote 12 down vote

Some other examples, consider datapoints that form a circle or ellipse, the covariance is 0, but knowing x you narrow y to 2 values. Or data in a square or rectangle. Also data that forms an X or a V or a ^ or < or > will all give covariance 0, but are not independent. If y = sin(x) (or cos) and x covers an integer multiple of periods then cov will equal 0, but knowing x you know y or at least |y| in the ellipse, x, <, and > cases.

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That if should be "if x covers an integer multiple of periods beginning at a peak or trough", or more generally: "If x covers an interval on which y is symmetric" – naught101 Feb 16 '12 at 0:47
up vote 11 down vote

Here is the example I always give to the students. Take a random variable $X$ with $EX=0$ and $EX^3=0$, e.g. normal random variable with zero mean. Take $Y=X^2$. It is clear that $X$ and $Y$ are related, but

$$cov(X,Y)=EXY-EX\cdot EY=EX^3=0.$$

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I like that example too. As a particular case, a N(0,1) rv and a chi2(1) rv are uncorrelated. – ocram Jul 15 '11 at 8:21
+1 but as a minor nitpick, you do need to assume that $E[X^3] = 0$ separately (it does not follow from the assumption of symmetry of the distribution or from $E[X] = 0$), so that we don't have issues such as $E[X^3]$ working out to be of the form $\infty - \infty$. And I am queasy about @ocram's assertion that "a N(0,1) rv and a chi2(1) rv are uncorrelated." (emphasis added) Yes, $X \sim N(0,1)$ and $X^2 \sim \chi^2(1)$ are uncorrelated, but not any $N(0,1)$ and $\chi^2(1)$ random variables. – Dilip Sarwate Feb 16 '12 at 3:09
@DilipSarwate, thanks, I've edited my answer accordingly. When I wrote it I though about normal variables, for them zero third moment follows from zero mean. – mpiktas Feb 16 '12 at 8:25
up vote 2 down vote

The first image on https://en.wikipedia.org/wiki/Correlation_and_dependence has a number of examples on the third row, in particular the first and the fourth example have a strong dependent relationship, but 0 correlation (and 0 covariance).

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up vote -2 down vote

Believe me I am not a statistician. But, I would like to respond because I am for some reasons, loking at it differetly. Generally, one can not assume that there is a zero relationship between two variables. Probably, zero relationship or association is hypothetical. Moreover, we look for covariation under the assumption that interaction between X and Y exceeds or does not exceed the underlying actual interaction ie mean * mean. The XY exceeds if we have moderators (or extraneous factors eg measurement error) operating. If moderator effects are not there, the covariance will always be zero. It does not mean that there is no correlation between the two variables!

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