Poisson Distribution

Poisson distributions are very important not only for counting events during a fixed period of time but also for different types of models, for instance, within credit risk management. In this post we focus on the most important features of a Poisson distribution.

We are interested in the following random variable

    \[ X' := \text{ \# events occurring in a fixed time period} \]

where \# is the number sign. An event can occur 0,1, \ldots times, i.e., the distribution range is \mathbb{N}_0 :=\mathbb{N} \cup \{0\}. The possible number of events is not restricted upwards, however, the possible natural numbers can occur with quite different probabilities.

To make things simpler we can, however, normalize the time frame within we count the events to the unit interval [0,1].

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Given the transformation of the time interval we are now interested in the following random variable:

    \[ X := \text{ \# events occurring in } [0,1] \]

However, the following four assumptions have to be fulfilled such that X is governed by the Poisson random variable:

  1. Two events can not occur at the same point in time;
  2. The probability for an event occurring during a short period of time with length \Delta t is approximately \lambda \Delta t as sketched in the figure above. If \Delta t is small enough, the corresponding probability will also decrease accordingly. The parameter \lambda is also called intensity parameter;
  3. The number of events occurring in two disjoint sub-intervals of [0,1] are independent. That is, knowing when one event happens under this circumstances provides no additional information about when another event will occur;
  4. The probability that an event occurs in a given length of time does not change through time. That is, the probability of occurrence only depends on the length but not on the position of the sub-interval.

If those assumptions hold true the random variable X is governed by the Poisson distribution.

A random variable X that follows the probability mass function

    \[ f(x) = \mathbb{P}(X=k) = \begin{cases} \frac{\lambda^k}{k!} \exp(-\lambda) & k \in \mathbb{N}_0\\ 0 & \text{else} \end{cases} \]

is distributed according to the Poisson distribution denoted by X \sim Pois(\lambda).

If X \sim B(n, p), i.e. if X follows a binomial distribution with parameters n\in \mathbb{N} and p\in [0,1], \mathbb{Q}(X=k) gives the probability of exactly k successes  in a sequence of n independent experiments , each asking a yes–no question, and each with its own boolean-valued outcome. What if we have an infinite number of trials and expect to see \lambda successes? This leads directly to the Poisson distribution:

(1)   \begin{align*} \lim_{n \rightarrow \infty} {\mathbb{Q} (X=k)}  &=  \lim_{n \rightarrow \infty} {\mathbb{Q} \left( \text{exactly $k$ successes in $n$  trials} \right)} \\ &=  \frac{\lambda^k}{k!} \exp(-\lambda) = \mathbb{P} (X=k). \end{align*}

What specific event we are interested in does not matter as long as the four assumptions above are (at least approximately) fulfilled. For instance, we could count the number of defaults occurring during a year,  the number of claims to an insurance company within a business year and so on.

The next exhibit, where different probability mass functions are plotted, shows that the probability of small values is greater if \lambda is smaller.

Several discrete probability mass function plots of Poisson distributed variables

The intensity parameter \lambda can also take on very small values. That is, if we choose \lambda small enough (e.g. \lambda < 0.01), it is very likely that we end up with a sample only containing \{0,1\}. Corresponding examples are plotted in the next graph.

Several discrete PDF plots of Poisson distributed variables with small \lambda

The cumulative distribution function of a Poisson-distributed random variable X is given by

    \[ F(k) = \mathbb{P}(X \leq k) = \exp(-\lambda) \sum_{i=0}^{\lfloor k\rfloor}{\frac{\lambda^i}{i!}}. \]

and sketched for \lambda \in \{5,10,20\} in the next exhibit:

Cumulative discrete distribution function for several parameter \lambda