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List every state
| X | Y = X² | Probability | XY |
|---|---|---|---|
| −1 | 1 | 1/3 | −1 |
| 0 | 0 | 1/3 | 0 |
| 1 | 1 | 1/3 | 1 |
Calculate the cross moment
E[X]=0, E[Y]=2/3 and E[XY]=0. Cov(X,Y)=E[XY]−E[X]E[Y]=0. Var(X)=2/3 and Var(Y)=2/9, so both standard deviations are positive and correlation is zero.
Check a conditional event
P(Y=0)=1/3. Given X=0, Y must be 0, so P(Y=0|X=0)=1. The changed conditional probability establishes dependence. In fact Y is determined by X in this model.
Keep the inference narrow
Covariance captures this second-moment relationship. Zero covariance does not generally rule out nonlinear dependence. The example does not supply a portfolio return forecast; it shows why a correlation label alone cannot describe every dependence risk.