The moving average model of order q has the form $$Y_t =β_0 +e_t +b_1e_{t−1} +b_2e_{t−2} +...+b_qe_{t−q}$$
where $e_t$ is a serially uncorrelated random variable with mean $0$ and variance $σ^2_e$.
(a) Show that $E(Y_t) = β_0$.
(b) Show that the variance of $Y_t$ is $\operatorname{var}(Y_t)=σ^2_e (1+b^2_1 +b^2_2 +...+b^2_q)$.
(c) Show that $ρ_j =0$ for $j > q$.
(d) Suppose that $q = 1$. Derive the autocovariances for $Y$.
I am unable to do part $c$ and $d$ even knowing the solutions. I have trouble calculating autocovariences in general so if someone could explain it, that'd be very helpful. Thank you!