*10*

**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 Î» 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}, â€¦, 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 Î».**

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

**Step 5: Set the derivative equal to zero and solve for Î».**

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

Thus, the MLE turns out to be:

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

**Additional Resources**

An Introduction to the Poisson Distribution

Poisson Distribution Calculator

How to Use the Poisson Distribution in Excel