Covariance of two plus-or-minus-one signals

Question

Two signals X and Y each take the value +1 or -1. Their joint chances are: Y=-1 Y=+1 --------- X=-1 0.30 0.20 X=+1 0.10 0.40 What is the covariance of X and Y?

Accepted Answer

Use the definition: \mathrm{Cov}(X, Y) = E[XY] - E[X]\,E[Y]. The averages. For X: it is +1 with chance 0.10 + 0.40 = 0.50 and -1 with chance 0.50, so E[X] = (+1)(0.50) + (-1)(0.50) = 0. Because E[X] = 0, the whole second term E[X]E[Y] vanishes, and the covariance is simply E[XY]. This is a handy shortcut whenever one variable is centered at zero. The product term. List each cell's product x \cdot y times its chance: E[XY] = (-1)(-1)(0.30) + (-1)(+1)(0.20) + (+1)(-1)(0.10) + (+1)(+1)(0.40). E[XY] = 0.30 - 0.20 - 0.10 + 0.40 = 0.40. Combine. \mathrm{Cov}(X, Y) = 0.40 - (0)\,E[Y] = 0.40. The pos…

Covariance of two plus-or-minus-one signals

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	$Y=-1$	$Y=+1$
$X=-1$	0.30	0.20
$X=+1$	0.10	0.40