Poisson Distribution#

PMF and CDF of Poisson Distribution#

Definition 38 (Poisson Distribution)

Let \(X\) be a Poisson random variable. Then the probability mass function (PMF) of \(X\) is given by

\[ \P \lsq X = k \rsq = e^{-\lambda} \cdot \frac{\lambda^k}{k!} \qquad \text{for } k = 0, 1, 2, \ldots \]

where \(\lambda > 0\) is called the Poisson parameter.

We write

\[ X \sim \poisson(\lambda) \]

to say that \(X\) is drawn from a Poisson distribution with parameter \(\lambda\).

Some conventions:

  1. The parameter \(\alpha\) determines the average rate of occurrences over time or space [Shea, 2021].

  2. \(T\) is a length of time/space being considered.

  3. \(\lambda = \alpha T\) is the average number of occurrences over the time/space being considered.

Plotting PMF and CDF of Poisson Distribution#

The below plot shows the PMF and its Empirical Histogram distribution for a Poisson distribution with \(\lambda \in \lset 5, 10, 20 \rset\).

1from plot import plot_poisson_pmfs, plot_empirical_poisson
2
3_fig, axes = plt.subplots(2, 1, figsize=(12, 10), dpi=125)
4lambdas = [5, 10, 20]
5plot_poisson_pmfs(lambdas=lambdas, ax=axes[0])
6plot_empirical_poisson(lambdas=lambdas, ax=axes[1])
7plt.show()

Using Seed Number 42
../_images/6433048c77820b6db30f7abffb010b62f18c080bbfbb2786a3ee2ad0fc71bb93.svg

Assumptions#

The Poisson distribution needs to satisfy the following assumptions before it can be used:

  1. An event can occur zero or more times in a fixed interval of time/space. In other words, \(k\) can be any non-negative integer.

  2. Events are independent of each other. In other words, the occurrence of one event does not affect the occurrence of another event.

  3. The average rate of occurrence of an event is constant over time/space. In other words, \(\lambda\) is constant and does not change over time/space.

  4. Two events cannot occur at the same instance in time/space. In other words, at very small time/space interval \(\Delta t\), either exactly one event occurs or no event occurs.

  5. The linearity assumption, the probability of an event occurring is proportional to the length of the time period. For example, it should be twice as likely for an event to occur in a 2 hour time period than it is for an event to occur in a 1 hour period1.

  6. The value of \(\lambda\) is proportional to the length of the time period.

Where did these Assumptions come from?

See chapter 3.5.4 on section Origin of the Poisson random variable of [Chan, 2021].

Example 10 (Example)

For example, the Poisson distribution is appropriate for modeling the number of phone calls an office would receive during the noon hour, if they know that they average \(4\) calls per hour during that time period.

  1. In the time period of one hour, the number of phone calls can be any non-negative integer. \(4\) just happens to be the average number of phone calls for that observed time period.

  2. The occurrence of one phone call does not affect the occurrence of another phone call. Caller A’s phone call does not affect the occurrence of caller B’s phone call in this time period.

  3. The average rate of occurrence of a phone call is constant over the time period. The average rate of occurrence of a phone call is \(4\) per hour. This may vary slightly but it is an assumption that we are making.

  4. The probability of more than one event occurring in a very small time period is near zero. It is unlikely that two phone calls will occur at the exact same time (ok maybe not but it is an assumption we are making).

  5. In an one hour time period, \(\P \lsq X = 2 \rsq = \frac{4^2}{2!} \cdot e^{-4} = 0.0183\). Then if we were to ask in a two hour time period, then \(\P \lsq X = 2 \rsq = 2 \cdot 0.0183 = 0.0366\). This is a linear assumption.

  6. For example, if the average rate of occurrence of an event is 1 per hour, then the average rate of occurrence of an event is 2 per 2 hours. Same idea as point 5.

Example 11 (Counter Example)

The number of students who arrive at the student union per minute will likely not follow a Poisson distribution, because the rate is not constant (low rate during class time, high rate between class times) and the arrivals of individual students are not independent (students tend to come in groups) [Wiki, 2022].

Properties#

Property 15 (Expectation of Poisson Distribution)

Let \(X \sim \poisson(\lambda)\) be a Poisson random variable with parameter \(\lambda\). Then the expectation of \(X\) is given by

\[ \expectation \lsq X \rsq = \sum_{k=0}^{\infty} k \cdot \P \lsq X = k \rsq = \lambda \]

Property 16 (Variance of Poisson Distribution)

Let \(X \sim \poisson(\lambda)\) be a Poisson random variable with parameter \(\lambda\). Then the variance of \(X\) is given by

\[ \var \lsq X \rsq = \expectation \lsq X^2 \rsq - \expectation \lsq X \rsq^2 = \lambda \]

Property 17 (Sum of Independent Poisson Random Variables)

Let \(X_1, X_2, \ldots, X_n\) be independent Poisson random variables with parameter \(\lambda_i\) for \(i \in \lset 1, 2, \ldots n \rset\). Then the sum of these \(n\) random variables is also a Poisson random variable with parameter \(\lambda = \sum_{i=1}^n \lambda_i\).

Poisson Approximation to Binomial Distribution#

Theorem 8 (Poisson Approximation to Binomial Distribution)

For situations where the number of trials \(n\) is large and the probability of success \(p\) is small, the Poisson distribution is a good approximation to the Binomial distribution.

Recall the Binomial distribution is given by

\[ X \sim \binomial(n, p) \qquad \text{with PMF} \qquad \P \lsq X = k \rsq = \binom{n}{k} p^k (1-p)^{n-k} \qquad \text{for } k = 0, 1, 2, \ldots, n \]

Then the approximation is given by

\[ \binom{n}{k} p^k (1-p)^{n-k} \approx e^{-\lambda} \cdot \frac{\lambda^k}{k!} !\qquad \text{for } k = 0, 1, 2, \ldots \]

where \(\lambda = np\).

Many times, Poisson PMF is much easier to compute than its Binomial counterpart.

Further Readings#

  • Chan, Stanley H. “Chapter 3.5.4. Poisson random variable.” In Introduction to Probability for Data Science, 152-164. Ann Arbor, Michigan: Michigan Publishing Services, 2021.

1

Be careful, this only holds in a small time interval, i.e. \(\P \lsq X(t + \Delta t) - X(t) = 1 \rsq = \lambda \Delta t\) for sufficiently small \(\Delta t\) [Chan, 2021].