Suppose that $Y1, Y2,...$ is an AR(1) process with $μ = 0.$5, $φ = 0.4$, and $σ^2 = 1.2$
(c) What is the variance of $(Y1 + Y2 + Y3)/2$?
I know that I need to do a linear transformation, but I'm not sure where to get started. Any thoughts?
I figured it out... $$Var[\frac{Y_1+Y_2+Y_3}{2}] = E[\frac{(Y_1+Y_2 +Y_3)^2}{2}] - E[\frac{(Y_1+Y_2+Y_3)}{2}]^2$$
$$= \frac{3}{2^2}\bigg[\sum_{n = 1}^{3}Var(Y_n) + 2 * Cov(Y_1,Y_2) + 2*Cov(Y_2,Y_3)\bigg]$$
This works out to be 2.48 for the above parameters.