Skip to content
Resources

    Covariance


    Covariance measures the degree of dependence between two random variables.  The covariance of two random variables X and Y is defined as follows:

    Cov(X, Y) = E(XYE(X) E(Y)

    Various properties of covariance include:
    • Cov(X, X) = Var(X)
    • Cov(X, Y) = Cov(Y, X)
    • Cov(aX, bY) = abCov(X, Y) for constants a and b
    • Cov(X+a, Y+b) = Cov(X, Y) for constants a and b
    Example:

    Determine the covariance between two random variables X and Y representing the numbers on the top and bottom of a fair die respectively.

    Solution:

    Using the fact that top and bottom numbers of dice always equal seven, in conjunction with the knowledge regarding a probability of 1/6 for each possible outcome for a fair die, it follows that:

    x y xy Pr(X=x) and Pr(Y=y)
    1 6 6 1/6
    2 5 10 1/6
    3 4 12 1/6
    4 3 12 1/6
    5 2 10 1/6
    6 1 6 1/6

    Using the entries above:

    E(X) = (1)(1/6) + (2)(1/6) + (3)(1/6) + (4)(1/6) + (5)(1/6) + (6)(1/6)

    E(X) = (1 + 2 + 3+ 4 + 5 + 6)/6

    E(X) = 21/6

    E(X) = 7/2 = 3.5


    E(Y) = (6)(1/6) + (5)(1/6) + (4)(1/6) + (3)(1/6) + (2)(1/6) + (1)(1/6)

    E(Y) = (6 + 5 + 4+ 3 + 2 + 1)/6

    E(Y) = 21/6

    E(Y) = 7/2 = 3.5


    E(XY) = (6)(1/6) + (10)(1/6) + (12)(1/6) + (12)(1/6) + (10)(1/6) + (6)(1/6)

    E(XY) = (6 + 10 + 12 + 12 + 10 + 6)/6

    E(XY) = 56/6

    E(XY) = 28/3 = 9.3333

    Employing Cov(X, Y) = E(XYE(X) E(Y):

    Cov(X, Y) = 9.3333  (3.5)(3.5)

    Cov(X, Y) = 9.3333  12.25

    Cov(X, Y) = 2.9167


    Related Topic: