# What implies correlation?

If I have three scalar random variables: $a$, $b$ and $c$, which are each zero-mean and have some nonzero variances, and I know:

1) The correlation between $a$ and $b$ is nonzero.

2) The correlation between $b$ and $c$ is nonzero.

Does this imply that the correlation between $a$ and $c$ is nonzero?

I feel like the answer must be yes, but I don't have any sound mathematical reasoning for it. Any advice would be greatly appreciated!

## Answers and Replies

D H
Staff Emeritus
Science Advisor
Simple counterexample: Suppose that a and c are uncorrelated random variables each with zero mean and nonzero variance and suppose that bac, with α and γ non-zero constants. By construction, b is correlated with each of a and c, but a and c are (by construction) uncorrelated.

Simple counterexample: Suppose that a and c are uncorrelated random variables each with zero mean and nonzero variance and suppose that bac, with α and γ non-zero constants. By construction, b is correlated with each of a and c, but a and c are (by construction) uncorrelated.

Many thanks for clearing that up so elegantly. It's easy when you know how!

Perhaps I can develop my understanding of a similar problem here without starting a new topic:

If, again, I have three scalar random variables $a$, $b$ and $c$ which are each zero-mean and have some nonzero variances... and in this case $a$ and $b$ are uncorrelated:

$\mathcal{E} \left\{ ab^*\right\} = 0$

where $\mathcal{E}\left\{\right\}$ denotes expectation and $*$ denotes complex conjugate (although the variables probably need not be complex for this example).

What I'd like to know is whether, in general, we can find a $c$ which can sort of 'recorrelate' $a$ and $b$:

$\mathcal{E}\left\{ cab^*\right\} > 0$

I can't seem to find such a case using numerical examples in Matlab, but I'd really like to figure out a proper mathematical approach to this. Any advice or insights would be very much appreciated!