Suppose X Y are IID Normalmu sigma2 random variables Find th

Suppose X, Y are IID Normal(mu, sigma2) random variables. Find the joint PDFs of the following Suppose X1, X2,X3 are IID random variables with PDF fx(x). Try finding the joint PDF of Y, = X2 - X1 Y2 = X3 - X2. What happens? Now find the joint PDF of Y1 = X2-X1 , Y2 = X3 - X2,Y3 = X3. Why does adding Y3 = X3 help?

Solution

1.a)

Z=X+Y~normal distribution with mean E[X+Y]=E[X]+E[Y]=2mu and variance =V[X+Y]=V[X]+V[Y] [as they are independent] =2sigma²

W=X

cov(W,Z)=cov(X,X+Y)=V(X)=sigma²   [as they are independent cov(x,y)=0]

so correlation coefficient=cov(W,Z)/sqrt(v(w)*v(z))=1/sqrt(2)=0.707

hence W and Z joint follows Bivariate normal with parameters (mu,sigma²,2mu,2sigma²,0.707) [answer]

b) as X and Y are identical hence here also like previous case W=Y and Z=X+Y joint follows Bivariate normal with parameters (mu,sigma²,2mu,2sigma²,0.707) [answer]