What implies correlation?

[weetabixharry]

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!

 

[D H]

Simple counterexample: Suppose that a and c are uncorrelated random variables each with zero mean and nonzero variance and suppose that b=αa+γc, 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.

 

[weetabixharry]

Simple counterexample: Suppose that a and c are uncorrelated random variables each with zero mean and nonzero variance and suppose that b=αa+γc, 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!

 

[weetabixharry]

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!

 

[blue_raver22]

Correlation refers to the degree of linear relationship between two variables. In other words, it measures how closely the values of two variables are related to each other. A nonzero correlation implies that there is a relationship between the two variables, and this relationship may be positive or negative.

In the given scenario, we have three variables - a, b, and c, and we know that the correlation between a and b is nonzero, and the correlation between b and c is nonzero. This means that there is a relationship between a and b, and between b and c.

Based on this information, it is reasonable to assume that there may also be a relationship between a and c. However, it is important to note that correlation does not necessarily imply causation. Just because there is a correlation between two variables does not mean that one variable causes the other.

To determine whether the correlation between a and c is nonzero, we would need to calculate the correlation coefficient between these two variables. If the correlation coefficient is nonzero, then we can say that there is a relationship between a and c. If the correlation coefficient is zero, then we cannot conclude that there is a relationship between a and c, even though there is a relationship between a and b and between b and c.

In conclusion, the given information suggests that there may be a relationship between a and c, but it cannot be confirmed without calculating the correlation coefficient between these two variables. It is important to always consider the limitations of correlation and to use caution when interpreting its results.