Let x and y be jointly distributed numeric variables and let

Let x and y be jointly distributed numeric variables and let z = a + by, where a and b
are constants. Show that cov(x, z) = b cov(x, y). Show that if b > 0, cor(x, z) = cor(x, y).
What happens if b < 0?

Solution

Given z= a+by

cov (x,z) = E(xz) - E(x) E(z) = E[x(a+by)] - E(x) E(a+by)

= E(ax+bxy) - E(x) {a+bE(y)}= aE(x) + bE(xy) - aE(x)-bE(x)E(y)

=bE(xy)-bE(x)E(y) = b[E(xy)-E(x)E(y)]=b cov(x,y)

if b<0 then z= a-by

cov (x,z) = E(xz) - E(x) E(z) = E[x(a-by)] - E(x) E(a-by)

= E(ax-bxy) - E(x) {a-bE(y)}= aE(x) - bE(xy) - aE(x)+bE(x)E(y)

=-bE(xy)+bE(x)E(y) =- b[E(xy)-E(x)E(y)] = -b cov(x,y)