Poisson distribution for arriving vehicles I'm interested to model the number of Electric Vehicles (EVs) which arrive to a charging station during one day and their Time-of-Arrivals (ToA).
I read that the number of EVs arriving at a charging station during a time interval is considered to follow a Poisson distribution, which uses a parameter which is called $\lambda$, which is determined by $λ=r \cdot t$, where:

*

*$r$ is the "arrival rate";

*$t$ is the "time duration".

Example:

*

*$r = 1 \space EV / hour$

*$t = 24 \space hours$
I know that the probability with which n = 20 EVs arrive at the charging station during 24 hours is:
$$ P(n=20) = \frac{e^{-r t} \cdot (r t)^n}{n!} = 0.0623 = 6.23 \% $$
but it's not what I'm looking for, because I'd like to obtain:

*

*number of Electric Vehicles (EVs) which arrive to a charging station during one day;

*and their Time-of-Arrivals (ToA);

or, alternatively (if what I request above is not possible), could also be sufficient to obtain:

*

*the number of EVs which arrives at each hour of the day.

Which could be a way to reach my goals?
 A: 
I'd like to obtain:

number of Electric Vehicles (EVs) which arrive to a charging station during one day;
and their Time-of-Arrivals (ToA);

Unfortunately the total number of arrivals in a day is not  deterministic quantity thus you cannot "know" it but you can have a probability information about how many they are. It is a random variable, governed by a Poisson process of parameter $\theta=24$ which means that, on average,  you will have 24 arrivals in a day (using your example)
The probability of any number of arrivals is given by the known formula
$$\mathbb{P}[X=x]=\frac{e^{-24}\cdot 24^x}{x!}$$
where $x=0;1;2;3;\dots$
For example
$$\mathbb{P}[14\leq X\leq 34]=96.9\%$$
this tells you that with a high level of confidence, the total number of arrivals will be in the interval $24\pm10$
As per the ToA is concerned, they follow a $\text{Gamma}[n;24]$ distribution.
A: *

*Number in a day is modeled by $\text{Poisson}\left(rt\right)$.


*Arrival times are modeled by $\text{Gamma}(n, r)$ with $n$ being the $n$th arrival.
