= Solution
A <right-censored> subject contributes survival up to the recorded censoring time, not no information. Under <independent censoring>, write $x_i=\min(T_i,C_i)$ and $v_i=\mathbf1_{\{T_i\leq C_i\}}$. For exponential death rate $\lambda$, the event-model <survival likelihood> is
$$
\boxed{L(\lambda)\propto\lambda^{\sum_i v_i}
e^{-\lambda\sum_i x_i},\qquad
\widehat\lambda_{\mathrm{full}}=\frac{\sum_i v_i}{\sum_i x_i}.}
$$
\b[Discarding censored subjects drops observed exposure and selects people whose death occurred before censoring.] Their recorded event times are not an unselected sample of all death times. The death-only conditional distribution can differ from the marginal death-time distribution even when censoring is independent. A correct <maximum-likelihood estimate> uses the total observed person-time, including censored follow-up.
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