Derive the mean and variance of X2 12X1 X2 Q3 X1 X2 X3 are

Derive the mean and variance of X2 = 1/2(X1 + X2). Q3. X1, X2, X3 are identically distributed random variables with the same mean and the same variance sigma^2 = gamma (0). The covariance between these random variables is gamma(Xj, Xi) = gamma(|i - j|), for i = 1, 2, 3 and j = 1, 2, 3.

Solution

cov (x1,x2) = E(x1x2)-E(X1)E(x2)

= E(x1x2) - mu^2

As they are identically distributed random variable

sigma^2 = gamma (0)

var (x1,x2) = gamma 1

Hence proved