I need to find the distribution of the random variable $T_{peak}$ where $T_{peak}$ represents the peak time of the first hitting time process.
Detailed Explanation of the System: There are $N^{Tx}$ amount of emitted molecules from a specific point in 3D environment. Molecules diffuse in the environment according to the followings:
$$ r[t] = r[t-1] + (\Delta r_1, \Delta r_2, \Delta r_3)$$ $$ \Delta r_i \sim \mathcal{N}(0,\, 2D\Delta t)$$ where $r[t]$, $r_i$, $D$, and $\Delta t$ are the location vector at time $t$, $i$-th component of the location vector, diffusion coefficient, and the time step, respectively.
If there is an absorbing spherical trap at a distance $d$, the mean number of arriving/hitting molecules until time $t$ is:
$$ E[N^{absorb}(t)] = N^{Tx} \frac{r_{trap}}{d+r_{trap}} \, \text{erfc} \left( \frac{d}{\sqrt{4Dt}} \right) = N^{Tx} \frac{r_{trap}}{d+r_{trap}} \, 2\Phi \left( \frac{-d}{\sqrt{2Dt}} \right)$$ where $r_{trap}$ is the radius of the absorbing spherical trap.
When you plot $N^{absorb}(t)$ in small intervals, you get something like the following figure (Scaled Inverse Gaussian distribution)
And the expected value of the peak time of hitting time histogram is $$ E[T_{peak}] = \frac{d^2}{6D}$$
When simulating this diffusion process and the focusing on the absorbing times, $T_{peak}$ differs from simulation instance to instance, $T_{peak}$ is a random variable and I need to find the distribution of $T_{peak}$.
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If we consider time bins $\{t_0, t_1, ..., t_k, ...\}$ and denote the number of absorbed molecules in that bin as $N(t_k)=N^{absorb}(t_k^-)-N^{absorb}(t_k^+)$ where $t_k^-$ and $t_k^+$ are the left and the right end of the $k$-th time bin, we can define $T_{peak}$ as follows: $$T_{peak} = \arg\max\limits_{t_k} N(t_k)$$ with this definition, what is the distribution of $T_{peak}$. Is it Poisson or Binomial or Gaussian or Inverse Gaussian? Does it have well known structure and name?
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P.S. These absorbed molecules are considered as the received signal in molecular communications and the peak time distribution of the received signal is important for many applications.