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Let's say we have three random variables $X, Y, Z$.

We know that $X$ and $Y$ are dependent, and also that $Y$ and $Z$ are dependent.

Under this setting, is it possible for $X$ and $Z$ to be independent?

(Intuitively, I think the answer is no. Knowing the value of $X$ would give us some information about $Y$, which in turn would give us some information of $Z$, making $X$ and $Z$ dependent. Although I can't seem to prove it.)

kjetil b halvorsen
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Dhruv Gupta
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    Intuitively, the part of Y that correlates with X can be different from the part of Y that correlates with Z, such that X and Z are independent of each. So this is very common. In fact, it is the situation in which ANCOVA is recommended; X is categorical, while Z and Y are continuous. – Heteroskedastic Jim Aug 07 '18 at 15:09
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    Imagine $X$ and $Z$ are the result of independent coin flips. Can you define $Y$ in such a way that it depends on both $X$ and $Z$? – Matthew Gunn Aug 07 '18 at 21:28

1 Answers1

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This is possible. Two examples:

Discrete example Throw two coins independently, but code the outcomes as 0 or 1. Then define:

$$ \begin{align} X&=\text{outcome on first coin}\\ Y&=\text{sum of both outcomes}\\ Z&=\text{outcome on second coin} \end{align} $$

Continuous variable example Let $(x,Y,Z)$ have the multivariate normal distribution $$ (X,Y,Z) \sim \mathcal{MN}(\begin{pmatrix}0\\0\\0\end{pmatrix},\begin{pmatrix}1&1/2&0\\1/2&1&1/2\\0&1/2&1\end{pmatrix} $$

Your arguments in your last paragraph can be formulated as a question? Is independence transitive? which leads to the analogous question Non-transitivity of correlation: correlations between gender and brain size and between brain size and IQ, but no correlation between gender and IQ

kjetil b halvorsen
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