Solution
= Solution
The lifetime $L$ is uniform on $[0,3]$ with mean $3/2$. The equilibrium age and excess densities are both
$$
\frac{\mathbb P(L>x)}{\mathbb EL}=\frac23\left(1-\frac x3\right).
$$
Hence, for $0\leq x\leq3$,
$$
\mathbb P(A_t\leq x),\ \mathbb P(E_t\leq x)
\longrightarrow\frac{2x}{3}-\frac{x^2}{9}.
$$
The observed total lifetime is length-biased, so
$$
\boxed{\mathbb P(A_t+E_t\leq x)
\longrightarrow\frac{\mathbb E[L\mathbf1_{L\leq x}]}{\mathbb EL}
=\frac{x^2}{9}.}
$$