# problem in understand exponential PDF [duplicate]

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

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?

• As you say, the formula is a density, not a probability. Please consult the paper for the meaning of their "$T.$" – whuber Nov 29 '19 at 20:32
• @whuber Would you please explain the differences, I'm really confused. – Pablo Nov 29 '19 at 20:35
• The duplicates contain such explanations. Search our site for threads that discuss probability and probability density for more. – whuber Nov 29 '19 at 20:40
• @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? – Pablo Nov 29 '19 at 22:22
• 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. – Glen_b -Reinstate Monica Nov 30 '19 at 7:33