7.1. Poisson Distribution#

We know that when $n$ is large and $p$ is small, the chance of $k$ successes in $n$ i.i.d. Bernoulli $(p)$ trials is roughly

where $\mu =np$.

The terms in the approximation are proportional to terms in the series expansion of $e^{\mu }$, but that expansion is infinite. It doesn’t stop at $n$, so we won’t either.

A little care is required before we go further. First, we must state the additivity axiom of probability theory in terms of countably many outcomes:

If events $A_1,A_2,\dots$ are mutually exclusive, then

This is called the countable additivity axiom, in contrast to the finite additivity axiom we have thus far assumed. It doesn’t follow from finite additivity, but of course finite additivity follows from it.

In this course, we will not go into the technical aspects of countable additivity and the existence of probability functions that satisfy the axioms on the spaces that interest us. But those technical aspects do have to be studied before you can develop a deeper understanding of probability theory. If you want to do that, a good start is to take Real Analysis and then Measure Theory.

While in this course, you don’t have to worry about it. Just assume that all our work is consistent with the axioms.

Here is our first infinite valued distribution.

7.1.2. An Interpretation of the Parameter#

To understand the parameter $\mu$ of the Poisson distribution, a first step is to notice that mode of the distribution is just around $\mu$. Here is an example where $\mu =3.74$. No computing system can calculate infinitely many probabilities, so we have just calculated the Poisson probabilities till the sum is close enough to 1 that the prob140 library considers it a Distribution object.

mu = 3.74
k = range(20)
poi_probs_374 = stats.poisson.pmf(k, mu)
poi_dist_374 = Table().values(k).probabilities(poi_probs_374)
Plot(poi_dist_374)
plt.title('Poisson (3.74)');

The mode is 3. To find a formula for the mode, follow the process we used for the binomial: calculate the consecutive odds ratios, notice that they are decreasing, and see where they cross 1. This is left to you as an exercise. Your calculations should conclude the following:

The mode of the Poisson distribution is the integer part of $\mu$. That is, the most likely value is $\mu$ rounded down to an integer. If $\mu$ is an integer, both $\mu$ and $\mu -1$ are modes.

mu = 4
k = range(20)
poi_probs_4 = stats.poisson.pmf(k, mu)
poi_dist_4 = Table().values(k).probabilities(poi_probs_4)
Plot(poi_dist_4)
plt.title('Poisson (4)');

In later chapters we will learn a lot more about the parameter $\mu$ of the Poisson distribution. For now, just keep in mind that the most likely value is essentially $\mu$.

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