# How come independence does not imply linear independence? (Statistics)

$$\begin{array}{c|lcr|c} \text{X/Y} & \text{0} & \text{100} & \text{200} &\text{Pr(X)}\\ \hline 100 & 0.2 & 0.1 & 0.2 & 0.5\\ 200 & 0.05 & 0.15 & 0.3 & 0.5\\ \hline \text {Pr(Y)} & 0.25 & 0.25 & 0.5 &1 \end{array}$$

Question: Are X and Y independent random variables? Can you say that X and Y are correlated or not? (X is row, Y is column).

The answer given says that X and Y are not independent but they might or might not be linearly independent therefore the correlation might or might not be zero.

How come we cannot imply there is no linear independence if the joint probability $\neq$ the product of marginal probabilities?

## 1 Answer

In general: if $X$ and $Y$ are independent, then they also are uncorrelated (zero correlation, or linear independence). The reverse is not true.

So, the title is wrong: independence does imply linear independence.

But the answer of the exercise does not say this, but the reverse, which is true. That is, $X$ and $Y$ could (in general) be uncorrelated (linear independent) even if they are not independent.

• I see, so independence implies linear independence but not independence does not imply linear independence. – user95087 Oct 21 '13 at 14:36
• "not independence does not imply not linear independence" – leonbloy Oct 21 '13 at 14:44