= Solution
In <discrete back-calculation with endpoint cohorts>, for equal-width intervals $I_k=[t_{k-1},t_k)$, write $\Delta=t_k-t_{k-1}$, $h_i$ for the expected infection count assigned to time $t_i$ under the end-of-interval approximation, and $\mu_k$ for the expected onset count in $I_k$. Define
$$
q_\ell=\Pr((\ell-1)\Delta\leq T<\ell\Delta),\qquad \ell=1,2,\ldots.
$$
An infection assigned to $t_i$ can first contribute to $I_{i+1}$, so \b[the discrete convolution with this timing convention is]
$$
\boxed{\mu_k\approx\sum_{i<k}h_iq_{k-i}.}
$$
The sum includes any infection cohorts before the observation window. With no infections before $t_0$ and cohorts beginning at $i=1$, it is $\sum_{i=1}^{k-1}$ and the first mean is zero in this endpoint approximation.
For unequal intervals use $q_{ik}=\Pr(t_{k-1}-t_i\leq T<t_k-t_i)$, truncating negative endpoints at zero, and $\mu_k\approx\sum_i h_iq_{ik}$. An alternative approximation assigning cohort $i$ to $t_{i-1}$ gives $\sum_{i\leq k}h_iq_{k-i+1}$. Both are legitimate discretisations; the subsequent solutions use the end-of-interval convention explicitly specified in part (d), so the index shift is essential.
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