Question

Let X1 and X2 be identically distributed random variables. Prove that Z1 = X1+X2 and Z2 = X1−X2 are uncorrelated. Are Z1 and Z2 independent?​

Answer 1

Answer:

Let X1 and X2 two independent normal distributions of parameters μ and σ2 and let Y1=X1+X2 and Y2=X1+2X2.

Compute the joint probability distribution of Y1 and Y2

This is what i tried

fY2|Y1(y2)fY1(y1)=fY1,Y2(y1,y2)

For a given x1+x2=y1, we have x1+2x2=y2⟺x2=y2−y1

Hence

fY2|Y1(y2)=fX2(y2−y1)

fY1,Y2(y1,y2)=fX2(y2−y1)fY1(y1)

Y1=X1+X2⟹Y1 is normal of parameters 2μ and 2σ2 and X2 is normal of parameters μ and σ2 so we can easily compute the product.