**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:

**Step 2: Write the likelihood function.**

Next, write the likelihood function. This is simply the product of the PDF for the observed values x_{1}, …, x_{n}.

**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:

Thus, the MLE turns out to be:

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

**Additional Resources**

An Introduction to the Poisson Distribution

Poisson Distribution Calculator

How to Use the Poisson Distribution in Excel

Awesome! Great help there.

simple and clear, thanks

Thanks for the post! I’m looking for a well-known reference (e.g., book or article) for the log-likelihood function of the Poisson distribution (I need to cite that reference in my paper). I was wondering if you could direct me to that reference. Thanks!