Solution
= Solution
Let
$$
\Lambda(t)=\int_0^t\lambda(s)\,ds
$$
be the baseline <cumulative hazard>. Under the <proportional hazards model>,
$$
h_i(t)=\lambda(t)e^{\beta^Tx_i},
\qquad
S_i(t)=\exp[-e^{\beta^Tx_i}\Lambda(t)].
$$
Thus
$$
f_i(Y_i)=h_i(Y_i)S_i(Y_i)
=\lambda(Y_i)e^{\beta^Tx_i}
\exp[-e^{\beta^Tx_i}\Lambda(Y_i)].
$$
Independence makes the log likelihood
$$
\boxed{\
\ell(\beta)=\sum_{i=1}^n
\left[
\log\lambda(Y_i)+\beta^Tx_i
-e^{\beta^Tx_i}\Lambda(Y_i)
\right]\
}.
$$