Let X(t) be a Gaussian process with mean µ_X (t) and autocovariance C_X(t,τ). In this problem, we verify that the for two samples X(t₁), X(t₂), the multivariate Gaussian density reduces to the bivariate Gaussian PDF. In the following steps, let σ²_i denote the variance of X(t_i) and let ρ = C_X (t₁,t₂ - t₁)/(σ₁σ₂) equal the correlation coefficient of X(t₁) and X(t₂).

(a) Find the covariance matrix C and show that the determinant is

(b) Show that the inverse of the correlation matrix is

(c) Now show that the multivariate Gaussian density for X(t₁), X(t₂) is the bivariate Gaussian density.