= Solution
For observation $j$ of bird $i$ in year $t(i,j)$, a <Gaussian linear mixed model> with crossed <random intercepts> is
$$
Y_{ij}=\beta_0+\beta_1 a_{ij}+u_i+v_{t(i,j)}+\varepsilon_{ij},
\quad
u_i\sim N(0,\tau_b^2),\quad v_t\sim N(0,\tau_y^2),\quad
\varepsilon_{ij}\sim N(0,\sigma^2),
$$
where the $u_i$, $v_t$, and $\varepsilon_{ij}$ are mutually <independent random variables>. Here $a_{ij}$ is age, $\beta_0$ is the population intercept at age zero, and $\beta_1$ is the change in conditional <expected value> per year of age. The $u_i$ capture persistent between-bird variation, the $v_t$ capture shared annual conditions, and the <regression residuals> capture remaining observation-level variation. Two observations from the same bird share <covariance> $\tau_b^2$; two from the same year share $\tau_y^2$, with both contributions present if both group labels agree. The model also assumes the <random effects> are independent of the age covariate.
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