# 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?

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