Correlated random-intercept and random-slope model ($Y_{ij}={\beta }_0+{\beta }_1{t}_{ij}+b_{0i}+b_{1i}{t}_{ij}+{\epsilon }_{ij}$)

A subject-specific random intercept and random slope may have a joint bivariate normal distribution with arbitrary positive-definite covariance matrix $D$. With independent normal measurement errors of variance ${\sigma }^2$, the marginal covariance between a person's readings at $s,t$ is $D_{00}+(s+t)D_{01}+st{D}_{11}$, plus ${\sigma }^2$ on the diagonal. Different subjects remain independent. Allowing $D_{01}$ distinguishes this model from an independent random-intercept and random-slope model.