Solution
Zero Correlation, Not Independence
Show the problem again
If two random variables have correlation exactly 0, are they necessarily independent? Answer yes or no, and have a counterexample ready.
Worked solution
No. Let X ~ Uniform(−1, 1) and Y = X². Then Cov(X, Y) = E[X³] − E[X]E[X²] = 0 − 0 = 0, yet Y is a deterministic function of X: maximally dependent. Correlation captures only linear association; independence requires the full joint distribution to factor. (The converse does hold: independence implies zero correlation.)
Source: Standard counterexample from mathematical statistics texts. Statement written for AxiomIQ.