Let X Y be Bivariate Normal with X and Y marginally N0 1 and

Let (X, Y) be Bivariate Normal, with X and Y marginally N(0, 1) and with correlation p between X and Y. Show that (X + Y. X - Y) is also Bivariate Normal. Find the joint PDF of X + Y and X - Y (without using calculus), assuming - 1

Solution

(X,Y) follows a bivariate normal with X and Y marginally N(0,1) with correlation coefficient p

then the MGF of X and Y is

M_x,y(t1,t2)=E[exp[t1X+t2Y]]=exp[t1²/2+t2²/2+pt1t2]    

let Z=X+Y and W=X-Y

then MGF of Z and W is

M_z,w(a,b)=E[exp[az+bw]]=E[exp[aX+aY+bX-bY]]=E[exp[(a+b)X+(a-b)Y]]=M_x,y(a+b,a-b)=mgf of bivariate normal

since MGF uniquely characterises a distribution. hence (Z=X+Y,W=X-Y) also follows a bivariate normal [proved]

b) now E[Z]=E[X+Y]=0

E[W]=E[X-Y]=0

V(Z)=V[X+Y]=V[X]+V[Y]+2cov(X,Y)=1+1+2p=2(1+p)

V(W)=V[X-Y]=V[X]+V[Y]-2cov(X,Y)=1+1-2p=2(1-p)

correlation coefficient is p_new=cov(X+Y,X-Y)/sqrt((2(1+p))*(2(1-p)))

now cov(X+Y,X-Y)=V(X)-V(Y)+cov(X,Y)-Cov(X,Y)=1-1=0

hence X and Y jointly follows a bivariate normal distribution with parameters (0,0,2(1+p),2(1-p),0)