Questions / Quant research, Probability & inference
Zero correlation is not independence
X is uniform on [−1, 1] and Y = X². What is Cov(X, Y)?
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Answer: 0 units²
- Cov(X, X²) = E[X³] − E[X]E[X²].
- By symmetry E[X³] = 0 and E[X] = 0.
- The covariance is 0, though Y is a function of X.
Interview shortcut. Odd moments vanish: Cov = 0.
Takeaway. Zero covariance is not independence.
In the app this has a follow-up where one thing about the setup changes. Try the follow-up.