MLE for a Poisson Distribution (Step-by-Step)

by Tutor Aspire January 17, 2023

Maximum likelihood estimation (MLE) is a method that can be used to estimate the parameters of a given distribution.

This tutorial explains how to calculate the MLE for the parameter λ of a Poisson distribution.

Step 1: Write the PDF.

First, write the probability density function of the Poisson distribution:

Poisson probability density function

Step 2: Write the likelihood function.

Next, write the likelihood function. This is simply the product of the PDF for the observed values x₁, …, x_n.

Likelihood function of Poisson distribution

Step 3: Write the natural log likelihood function.

To simplify the calculations, we can write the natural log likelihood function:

Step 4: Calculate the derivative of the natural log likelihood function with respect to λ.

Next, we can calculate the derivative of the natural log likelihood function with respect to the parameter λ:

Step 5: Set the derivative equal to zero and solve for λ.

Lastly, we set the derivative in the previous step equal to zero and simply solve for λ:

MLE of Poisson distribution

Thus, the MLE turns out to be:

Maximum likelihood estimation of Poisson distribution

This is equivalent to the sample mean of the n observations in the sample.