### Find a family of random variables for which zero correlation implies independence

Apart from the family of bivariate normal distributions.

- Created by Admin, Sep 27'22

Suppose [math]X[/math] and [math]Y[/math] are independent random variables with variances equal to 1. Then consider the family of random variables [math]aX + bY[/math] where [math]a[/math] and [math]b[/math] are non-negative. If [math]W = aX + bY[/math] and [math]Z = cX + dY[/math] are two non-zero uncorrelated members of this family then

[[math]]\operatorname{Cov}(W,Z) = ac + bd = 0.[[/math]]

But this implies that [math]ac = 0 [/math] and [math]bd = 0 [/math] which in turn implies that [math]W = aX [/math] and [math]Z = dY[/math] or [math]W = bY[/math] and [math]Z = cX [/math]. Hence uncorrelated members of the family are independent.

- Created by Admin, Sep 27'22