Question

Quality-control checks on wood paneling involve counting the number of surface flaws on each panel. On a given 2 x 8 ft panel, lel be the number of surface flaws due to uneven application of the final coat of finishing material, and lefV be the number of surface flaws due to inclusions of foreign particles in the finish. The joint probability mass function p (x,y) ofX and Y is presented in the following table. The marginal probability mass functions are presented as well, in the margins of the table. Find the covariance al and Y. 0.05 0.05 0.25 Pv)0.35 0.10 0.10 0.05 0.25 0.20 0.10 0.10 0.40 0.35 0.25 0.40 Cov(X, Y) =

Answer 1

The covariance is computed as:

Cov (X, Y) = E(XY) - E(X)E(Y)

We compute the various terms as:
E(XY) = 1*1*0.1 + 1*2*0.1 + 2*1*0.05 + 2*2*0.1 = 0.1 + 0.2 + 0.1 + 0.4 = 0.8

E(X) = 0*0.35 + 1*0.25 + 2*0.4 = 1.05
E(Y) = 0*0.35 + 1*0.25 + 2*0.4 = 1.05

Therefore, the covariance here is computed as:

Cov(X, Y) = 0.8 - 1.05*1.05 = -0.3025

Therefore the required covariance here is - 0.3025