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I am looking for the name of the following phenomenon. There are three random variables, $X,Y,Z$. We have $P(X,Y) \neq P(X)P(Y)$ and $P(Y,Z) \neq P(Y)P(Z)$. In other words, $X$ and $Y$ are dependent, and $Y$ and $Z$ are dependent.

Now, $X$ and $Z$ are dependent, through their mutual dependence on $Y$. I am interested in this phenomenon and how it pertains to expected values of products. For example, knowing $E[XY]$ and $E[YZ]$ I should be able to compute $E[XZ]$.

I have seen this called "cross correlation" which I think is wrong. Also, it seems different than "conditional dependence." But it must be very common and so I am asking for help in pinning this down.

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    $\begingroup$ First, it does not follow that $(X,Z)$ are not independent. For examples, you could start with any independent $(X,Z)$ and throw in any variable $Y$ that is dependent on both $X$ and $Z.$ This (strongly) suggests your aim is hopeless: you should not expect to compute $E[XZ]$ from the other two product expectations. In fact, if you restrict to the case where all the variables are standardized (zero mean, unit variance), these are indeed the Pearson correlations and your question has thereby been reduced to stats.stackexchange.com/questions/72790, which has good answers. $\endgroup$
    – whuber ♦
    Commented Jun 23 at 15:50
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    $\begingroup$ Thanks for this. It appears that there may not be a name for this because it is not a general property as I had assumed. In the case that both $X,Z$ actually depend on $Y$, however, my supposition would hold. But I suppose this is a special case which has no specific reason to be named. $\endgroup$
    – Wapiti
    Commented Jun 24 at 0:39

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