Covariance of buying coffee and buying a pastry

Question

Let X = 1 if a customer buys coffee (else 0) and Y = 1 if they buy a pastry (else 0). The shop's data gives the joint chances of each combination: Y=0 Y=1 --------- X=0 0.40 0.10 X=1 0.20 0.30 What is the covariance of X and Y?

Accepted Answer

Covariance is a single number that says whether two quantities tend to be high together or not. Positive means they rise and fall together; negative means one tends to be high when the other is low; zero means no straight-line link. The formula is \mathrm{Cov}(X, Y) = E[XY] - E[X]\,E[Y], which in words says: the average of the product, minus the product of the averages. The two averages. Since X and Y are 0 or 1, their averages are just the chances of being 1. Add across each row and column: E[X] = P(X=1) = 0.20 + 0.30 = 0.50, \qquad E[Y] = P(Y=1) = 0.10 + 0.30 = 0.40. The average of the prod…

Covariance of buying coffee and buying a pastry

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	$Y=0$	$Y=1$
$X=0$	0.40	0.10
$X=1$	0.20	0.30