Maximum value of a Poisson Distribution

Jegor, a subscriber to the Intmath Newsletter, recently asked this question:

When I was going through Probability Theory in Interactive Mathematics, I found Poisson Distribution quite interesting.

When I tried to find the maximum Probability using differentiation I suddenly realized that I don't know how to deal with differentiation when there is the Factorial in formula. Internet search gave me quite complicated solutions so I left it behind for now. If you find this problem interesting then it can be an interesting topic for the newsletter.

Well, I agree it's an interesting problem, so here we go. First, some background on what Jegor is asking.

Poisson Distribution - an Overview

Whenever there is a court case involving rare diseases (like Parkinson's Disease (which I wrote about in Michael J. Fox and the Parkinson’s cluster) or cancer (Probability of a cancer cluster: 1 in a million), the prosecution and defense argue over their interpretations of the Poisson Distribution. [See the whole thing here: Poisson Distribution.]

Poisson Distribution only applies if the events in question are independent. We also need to count the number of "successes" (or failures), so the variables involved need to be non-negative integers.

The formula for the probability distribution of a Poisson random variable X is given by the formula

where

Typically, this will produce a distribution as follows.

Let's take Example 2 from my Poisson Distribution page. In that example, we had a table indicating the number of flaws found in a manufacturing process and we needed to determine the probabilities of finding flaws in future manufacturing runs.

The average (mean) was μ = 2.3 and after applying the above formula for x = 0, x = 1, x = 2, etc, we obtained this histogram (bar chart, not a continuous curve) for the probability.

Poisson Distribution for Continuous Variables

By definition, the x in our Poisson formula is discrete, since we need to count the number of flaws (or sickness cases, or whatever).

So we can't actually find the derivative of the Poisson Distribution, since differentiation only works for continuous functions. Also, x! (factorial x) only has meaning if x is a non-negative integer.

However, as a hypothetical exercise, it could be interesting to proceed.

In an earlier article, I wrote about how the Gamma Function gives us continuous values for x!. That is, we can find some meaning for expressions like 3.5!.

Now, to find the derivative of the formula,

we need to be able to find the derivative of x! (that is, "x factorial").

Recall the following Gamma Function, which is analogous to x!:

The derivative of the Gamma Function is defined as:

And what is that Psi (Ψ)function in there? It is defined as the "logarithmic derivative of the gamma function"! This is rather a circular definition, and not very helpful, so let's see what we can figure out using graphs.

The Graph of the Poisson Function

Let's continue to assume we have a continuous variable x and graph the Poisson Distribution,

It will be a continuous curve, as follows.

I have included the (valid) discrete case (the black rectangles) for comparison. (You can graph this yourself using Wolfram | Alpha. You can enter it as follows (e^-2.3)(2.3^x)/x! from 0 to 10.)

Our aim is to find the value of x that gives maximum value of the probability P(X). We can see it is slightly less than x = 2.

We can graph the derivative of the Poisson distribution using a computer algebra system, as follows. The value we need will be given by the X-axis intercept.

(You can also get Wolfram \ Alpha to do this for you:

d/dx((e^-2.3)(2.3^x)/x!) from 0 to 3

Zooming in on the portion near x = 2, we obtain:

From the above, we can see the maximum will occur when x = 1.78223.

Substituting this value into the Poisson expression gives us the maximum value:

 

Conclusion

Our result tallies nicely with the graph of the "continuous" Poisson Distribution above. That is, we have successfully found the "maximum". We did it using graphical means, since an algebraic approach appears rather obscure.

However, while we have found a hypothetical "maximum" value of the Poisson Distribution for this example, it is not valid since Poisson Distribution only applies for discrete, countable variables.

See the 9 Comments below.