A negative covariance between taking the stairs and the elevator

Question

On any given trip in a building, let X = 1 if a person takes the stairs and Y = 1 if they take the elevator (each is 0 otherwise). The joint chances are: Y=0 Y=1 --------- X=0 0.10 0.40 X=1 0.40 0.10 What is the covariance of X and Y?

Accepted Answer

Use the definition, which compares the average of the product with the product of the averages: \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] = 0.40 + 0.10 = 0.50, \qquad E[Y] = 0.40 + 0.10 = 0.50. The product term. XY = 1 only when both are 1: E[XY] = P(X=1, Y=1) = 0.10. Combine. \mathrm{Cov}(X, Y) = 0.10 - (0.50)(0.50) = 0.10 - 0.25 = -0.15. The covariance is negative, which fits the story: taking the stairs and taking the elevator trade off, so being high on one goes with being low on the other. Takeaway. A negative cova…

A negative covariance between taking the stairs and the elevator

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More Probability questions

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