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Specify a complete finite model
Let X take −1, 0 and 1 with equal probabilities, and let Y = X². This is an original illustrative probability model, not a return forecast.
| X | Y = X² | Probability |
|---|---|---|
| −1 | 1 | 1/3 |
| 0 | 0 | 1/3 |
| 1 | 1 | 1/3 |
Calculate the covariance
E(X) = 0, E(Y) = 2/3 and E(XY) = E(X³) = 0. Covariance is E(XY) − E(X)E(Y), so it is zero. Both variables have positive variance: Var(X) = 2/3 and Var(Y) = 2/9. Their Pearson correlation is therefore zero with a valid positive denominator.
Test a conditional probability
Before observing Y, P(X = 0) = 1/3. Given Y = 0, the model requires X = 0, so P(X = 0|Y = 0) = 1. The information changes the probability, establishing dependence despite zero correlation.
Avoid turning a linear statistic into a general relationship test
The negative and positive covariance contributions cancel in this symmetric model. The non-linear mapping remains exact. Use the model’s joint distribution or appropriate dependence analysis when the question is broader than linear co-movement.