= Solution
The given <probability density function> is the completion-time density for two independent sequential irreversible steps, with <exponential distributions> of waiting times at rates $k_a,k_b>0$. Let $T=T_a+T_b$. The <convolution> is
$$
p_T(t)=\int_0^t k_ae^{-k_au}k_be^{-k_b(t-u)}\,du
=\boxed{\frac{k_ak_b}{k_a-k_b}(e^{-k_bt}-e^{-k_at}),\quad t\geq0.}
$$
It is a two-stage <hypoexponential distribution>. The density vanishes at zero, initially rises as $k_ak_bt$, peaks, and then decays with a tail governed by the slower rate. Setting its <derivative> to zero gives
$$
\boxed{t_{\rm mode}=\frac{\ln(k_a/k_b)}{k_a-k_b},\qquad
\langle T\rangle=\frac1{k_a}+\frac1{k_b},\qquad
\operatorname{Var}(T)=\frac1{k_a^2}+\frac1{k_b^2}.}
$$
Normalization follows either from the <convolution> of normalized densities or directly from $(k_ak_b/(k_a-k_b))(1/k_b-1/k_a)=1$. In the equal-rate limit $k_a=k_b=k$, the apparent singularity is removable and
$$
\boxed{p_T(t)=k^2t e^{-kt},\qquad t_{\rm mode}=1/k,\quad \langle T\rangle=2/k.}
$$
This is an <Erlang distribution> with shape two. If one rate is much faster, its short waiting time becomes hard to resolve and the distribution resembles a single <exponential distribution> after its initial rise.
The observation supports at least two kinetically resolved stages before the recorded event, rather than one memoryless rate-limiting event. Possibilities include substrate binding followed by a conformational or catalytic step. It does not identify which stage is binding, chemistry, product release or rotation. The density is unchanged by exchanging $k_a$ and $k_b$, so temporal order and chemical identity cannot be recovered from this fit alone. Other hidden-state mechanisms can also mimic the same density. Dependence of one fitted rate on <ATP> <concentration>, for example, would provide additional evidence about assignment, rather than following from the distribution by itself.
\Image[/past-exam-of-the-biological-physics-course-of-the-university-of-oxford/2021-channel-and-dwell-kinetics.png]
{title=Computed <two-state ion-channel gating> and two-stage waiting-time densities}
The left panel illustrates <1/b> for an initially closed population. The other panels are dimensionless examples of the distributions derived here; none reconstructs the missing source trace.
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