The Poisson process is a continuous process, and there can be multiple interpretations of it, which lead to different possible definitions. In this section, we will start with the formal definition and build up to a more simple, intuitive definition. A continuous-time stochastic process N(t):t > 0 is a Poisson process with a rate λ > 0 if the following conditions are met:
- N(0) = 0
- It has stationary and independent increments
- The distribution of N(t) is Poisson with mean λt:
First of all, we need to define what the stationary and independent increments are. For a continuous-time stochastic process, X(t): ≥ 0, an increment is defined as the difference in state of the system between two time instances; that is, given two time instances s and t with s < t, the increment from time s to time t is X(t) - X(s). As the name suggests, a process is said to have a stationary increment if its distribution for the increment depends only on the time difference.
In other words, a process is said to have a stationary increment if the distribution of X(t1) - X(s1) is equal to X(t2) - X(s2) if t1 > s1,t2 > s2 and t1 - s1 = t2 -s2. A process is said to have an independent increment if any two increments in disjointed time intervals are independent; that is, if t1 > s1 > t2 > s2, then the increments X(t2) - X(s2) and X(t1) - X(s1) are independent.
Now let's come back to defining the Poisson process. The Poisson process is essentially a counting process that counts the number of events that have occurred before time t. This count of the number of events before time t is given by N(t), and, similarly, the number of events occurring between time intervals t and t + s is given by N(t + s) - N(t). The value N(t + s) - N(t) is Poisson-distributed with a mean λs. We can see that the Poisson process has stationary increments in fixed time intervals, but as 
, the value of N(t) will also approach infinity; that is, 
. Another thing worth noting is that, as the value of λ increases, the number of events happening will also increase, and that is why λ is also known as the rate of the process.
This brings us to our second simplified definition of the Poisson process. A continuous-time stochastic process N(t): t ≥ 0 is called a Poisson process with the rate of learning λ > 0 if the following conditions are met:
- N(0) = 0
- It is a counting process; that is, N(T) gives the count of the number of events that have occurred before time t
- The times between the events are distributed independently and identically, with an exponential distribution having a learning rate of λ