Suppose that, given a vector of random effects, , observations y1, . . . , yn are

Suppose that, given a vector of random effects, α, observations y1, . . . , yn are (conditionally) independent such that yi ∼ N(x



iβ + z



iα, τ 2), where xi and zi are known vectors, β is an unknown vector of regression coefficients, and τ 2 is an unknown variance. Furthermore, suppose that α is multivariate normal with mean 0 and covariance matrix G, which depends on a vector

θ of unknown variance components. Let X and Z be the matrices whose ith rows are x



i and z



i , respectively. Show that the vector of observations, y = (y1, . . . , yn)

, has the same distribution as the Gaussian linear mixed model (1.1), where α ∼ N(0,G),  ∼ N(0, τ2I), and α and  are independent.