Simulating a Poisson RV

The Poisson distribution is a discrete probability distribution, which describes the number of times an event occurs during a specific interval; which can be of time, distance, area, volume, among others. For example, Poisson variables are: the number of calls received by a telephone exchange in a period of 1 minute, the number of bacteria in a volume of 1 liter of water or the number of faults on the surface of a rectangular piece of ceramic.

e.g. If we expect 4 patients to arrive every day, is it possible to calculate the probability of 3 arriving in a day, or 5 patients in a day? Yes, Poisson.

#Poisson Distribution

In a Poisson distribution, the following assumptions are always met:

• The random variable is the number of times an event occurs during a defined interval. The interval can be time, distance, area, volume, etc. X = number of times an event occurs during a defined interval.
• The probability of occurrence is the same for any 2 intervals of equal length.
• The occurrence or non-occurrence in any interval is independent of the occurence or non-occurrence in any other interval.
• Two events cannot occur at exactly the same time.

If these conditions are met, the discrete random variable X follows a Poisson distribution and will have the pmf:

where:

• f(x) = P(X = x) = p(x): probability of x occurrences in an interval
• $\lambda = E(X)=V(X)$: mean number of events within a given interval
• e: constant, euler number, base for natural logarithms = 2.71828
• The sample space of a Poisson RV is the set of all non-negative numbers

  

  The interesting thing about Poisson variables is that we can modify (if the model allows it) the time interval (0, t] in which we count the events. Of course, this does not necessarily have to be possible. But in general if the variable is poisson in (0, t] it will be also in any sub-interval (0, t’] for all t’ such that 0 < t’ < t. So we will be able to define a series of variables $X_t$ of distribution Po(t).

#Poisson Process Definition

Let’s consider a Poisson experiment with λ equal to the average number of events in a unit of time. If t is an amount of time in time units, the random variable $X_t$ = number of events in the interval (0,t] is a Po(λ⋅t). The set of variables ${X_t}_{t \gt 0}$ is called a Poisson process. Python provides a straightforward implementation of the Poisson distribution through the stats library, using the functions dpois for the pmf and ppois for the cdf. They require two arguments:

• first argument is the value at which to compute the pmf or the cdf;
• second is the parameter λ of the Poisson;

So for instance poisson.pmf returns P(X=3) = p(3) for a Poisson random variable with parameter λ = 1.

poisson.pmf(3,1) #returns 0.06131324

Returns P(X ≤ 8) = F(8) for a Poisson random variable with parameter λ = 4.

poisson.cdf(8,4) #return 0.9786366

#Random numbers from a Poisson rpois

We can also generate pseudo-random numbers from a poisson distribution using the np.random.possion(lambda, n). Arguments:

• lambda: the expectation for a specific time interval.
• n: amount of pseudo-random numbers

np.random.poisson(4,100)

#returns
array([ 3, 2, 0, 4, 5, 6, 2, 0, 5, 7, 5, 3, 4, 1, 4, 3, 1,
3, 11, 6, 5, 2, 3, 2, 6, 2, 6, 3, 2, 1, 5, 5, 2, 1,
3, 4, 6, 4, 3, 7, 1, 3, 3, 3, 1, 3, 1, 4, 4, 3, 4,
7, 1, 2, 3, 1, 3, 4, 2, 4, 3, 3, 4, 6, 2, 3, 7, 2,
5, 5, 8, 4, 5, 4, 4, 10, 2, 3, 6, 4, 2, 2, 6, 3, 6,
4, 5, 3, 7, 3, 3, 2, 3, 5, 3, 2, 6, 7, 6, 4])

Returns a sequence of 100 observations for a Poisson distribution of λ = 4.