I'm studying a paper called "Optimization based on bacterial chemotaxis". As it can be understood from its name, it has proposed an optimization algorithm based on the reaction of a bacterium toward its environment. The bacterium tends to go towards chemoattractants and stay away from repellents. The path of the bacterium consists of a sequence of straight-line trajectories that each of them has a different direction and duration.
In order to compute the duration of each trajectory, it has been written in the paper that:
Compute the duration of the trajectory $τ$ from the distribution of a random variable with an exponential probability density function:
$$P(X=τ) = 1/T e^{-τ/T}$$
where for $f_pr / l_pr >= 0$, $T=T_0$ and for $f_pr / l_pr < 0$, $T=T_0 (1+b\mid f_pr / l_pr|)$
I'm new to this field, but according to what I've studied here, my question is that since the duration is computed by exponential PDF, shouldn't they use $f(τ)$ instead of $P(X=τ)$? If not, what does $P(X=τ)$ mean here?