Covariance of rain and a food truck selling out

Question

On a given day, let X = 1 if it rains and Y = 1 if a food truck sells out (each is 0 otherwise). The joint chances are: Y=0 Y=1 --------- X=0 0.50 0.10 X=1 0.10 0.30 What is the covariance of X and Y?

Accepted Answer

Apply the definition: \mathrm{Cov}(X, Y) = E[XY] - E[X]\,E[Y]. The two averages. For 0/1 variables the average is the chance of a 1: E[X] = 0.10 + 0.30 = 0.40, \qquad E[Y] = 0.10 + 0.30 = 0.40. The product term. XY = 1 only in the bottom-right cell: E[XY] = P(X=1, Y=1) = 0.30. Combine. \mathrm{Cov}(X, Y) = 0.30 - (0.40)(0.40) = 0.30 - 0.16 = 0.14. The positive covariance says rainy days and sell-out days tend to occur together more often than chance alone would suggest. Takeaway. The whole computation for two indicators is one subtraction: the chance both events land together minus the produc…

Covariance of rain and a food truck selling out

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	$Y=0$	$Y=1$
$X=0$	0.50	0.10
$X=1$	0.10	0.30