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 calculate the duration of each trajectory (denoted by τ), they have used an exponential probability density function like this:

$$P(X=τ) = 1/T e^{-τ/T}$$

where for $f_pr / l_pr >= 0$, $T=T_0$ and for $f_pr / l_pr < 0$, $T=T_0 (1+b|f_pr / l_pr|)$

I'm new to exponential distribution, as far as I know the formula for the exponential distribution is:

$P(X>x) = e^{-\lambda x}$ and $P(X<x) = 1-e^{-\lambda x}$

and the probability of the value of $P(X= a.constant.number)$ is equal to $0$, isn't it? If so, what is the meaning of the presented formula in the paper?

My second question is that: according to what I stated about the paper, does $T$ here denote the mean duration of the trajectories up to this moment?

  • $\begingroup$ As you say, the formula is a density, not a probability. Please consult the paper for the meaning of their "$T.$" $\endgroup$ – whuber Nov 29 '19 at 20:32
  • $\begingroup$ @whuber Would you please explain the differences, I'm really confused. $\endgroup$ – Pablo Nov 29 '19 at 20:35
  • $\begingroup$ The duplicates contain such explanations. Search our site for threads that discuss probability and probability density for more. $\endgroup$ – whuber Nov 29 '19 at 20:40
  • $\begingroup$ @whuber Since the formula presented in paper is the exponential probability density function, is it correct to use $P(X=τ)$ to show the density function? Shouldn't it be f(x) instead? $\endgroup$ – Pablo Nov 29 '19 at 22:22
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    $\begingroup$ It is not correct to say that the density is $P(X=\tau)$ (this is explained in the indicated duplicates). If the paper said that, they have made an error. $\endgroup$ – Glen_b Nov 30 '19 at 7:33