Two variates, x and y , are uncorrelated and have standard deviations _x and _y respectively. What is the correlation coefficient between x + y and x - y ?

Step-by-step solution

Let u=(x+y) ; v=(x-y) aligned & u =( x + y ) ; v =( x - y ) & cov (u, v)=E (u- u )(v- v ) & =E (x- x )+(y- y ) (x- x )-(y- y ) & =E (x- x )^2-(y- y )^2 = _x^2- _y^2 & var (u)=E(u- u )^2=E (x- x )+(y- y ) ^2= _x^2+ _y^2 aligned Therefore x and y are uncorrelated. E(x- x )(y- y )=0 Similarly, var (v)= _x^2+ _y^2 Thus, r(u, v)= cov (u, v) var (u) var (v) = _x^2- _y^2 _x^2+ _y^2