The Poisson distribution models the probability of a number of events within a given time interval, assuming that these events occur with a known average rate and successive events occur independently of the time that has passed since the previous event.

A concrete example of a process that can be modeled by a Poisson distribution would be if an individual received an average of, say, 23 emails per day. If we assume that the arrival times for the emails are independent of each other, then the total number of emails an individual receives each day can be modeled by a Poisson distribution.

Another example could be the number of trains that stop at a particular station each hour. The PMF for a Poisson distribution is given by the following expression:

 

Here, λ is the rate parameter, which, represents the expected number of events/arrivals that occur per unit time, and k is the random variable that represents the number of events/arrivals.

The expectation and variance are given respectively by the following expressions:

For more information, refer to http://en.wikipedia.org/wiki/Poisson_process.

The PMF is plotted using matplotlib for various values as follows:

    In [11]: %matplotlib inline
       import numpy as np
       import matplotlib
       import matplotlib.pyplot as plt
       from scipy.stats import poisson
       colors = matplotlib.rcParams['axes.color_cycle']
       k=np.arange(15)
       plt.figure(figsize=(12,8))
       for i, lambda_ in enumerate([1,2,4,6]):
            plt.plot(k, poisson.pmf(k, lambda_), '-o',  
            label="$lambda$=" + str(lambda_), color=colors[i])
            plt.legend()
       plt.title("Possion distribution PMF for various $lambda$")
       plt.ylabel("PMF at $k$")
       plt.xlabel("$k$")
       plt.show()
  

The following is the output: