Suppose that two random variables X1 and X2 have a bivariate normal distribution, and Var(X1) = Var(X2). Show that the sum X1+X2 and the difference X1− X2 are independent.

Answer 1

Solution:
Let μ1 and μ2 be the means of X1 and X2 respectively, and σ1 and σ2 be the standard deviations of X1 and X2 respectively. Since Var(X1) = Var(X2), we have σ1 = σ2.

The joint probability density function of X1 and X2 is given by:
f(x1,x2) = (1/2πσ1σ2)exp[−(x1−μ1)2/2σ12 − (x2−μ2)2/2σ22]

The probability density function of X1+X2 is given by:
f(x1+x2) = (1/2πσ1σ2)exp[−(x1+x2−μ1−μ2)2/2σ12 − (x1+x2−μ1−μ2)2/2σ22]

The probability density function of X1−X2 is given by:
f(x1−x2) = (1/2πσ1σ2)exp[−(x1−x2−μ1+μ2)2/2σ12 − (x1−x2−μ1+μ2)2/2σ22]

Since the probability density functions of X1+X2 and X1−X2 are independent of each other, X1+X2 and X1−X2 are independent.