Correlation built from raw averages and cross-products

Question

For two quantities X and Y you are told the following averages: E[X] = 2, \quad E[X^2] = 8, \quad E[Y] = 3, \quad E[Y^2] = 18, \quad E[XY] = 9. What is the correlation between X and Y?

Accepted Answer

Everything comes from three building blocks: each variance and the covariance, each of which is an "average of the product minus product of the averages" computation. The two variances. \mathrm{Var}(X) = E[X^2] - (E[X])^2 = 8 - 2^2 = 4, \qquad \mathrm{SD}(X) = 2. \mathrm{Var}(Y) = E[Y^2] - (E[Y])^2 = 18 - 3^2 = 9, \qquad \mathrm{SD}(Y) = 3. The covariance. \mathrm{Cov}(X, Y) = E[XY] - E[X]\,E[Y] = 9 - (2)(3) = 3. The correlation. Divide the covariance by the two standard deviations: ho = \frac{3}{2 	imes 3} = \frac{3}{6} = 0.5. Takeaway. Variance is just the covariance of a variable with i…

Correlation built from raw averages and cross-products

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