How to Find the Maximum Likelihood Estimator (MLE) of the Standard Deviation in a Poisson Distribution

Understanding the Maximum Likelihood Estimator (MLE) for the standard deviation of a Poisson distribution can seem daunting, especially when dealing with an unknown mean. In this article, we will walk through the process step-by-step, from the basics of the Poisson distribution to the derivation of the MLE for the standard deviation. We'll also explore the likelihood and log-likelihood functions to help you grasp the concept more effectively.

Poisson Distribution Basics

The Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space. It is characterized by a single parameter, the mean (λ), denoted as lambda:

If X follows a Poisson distribution with mean λ, then the probability mass function is given by:

where x 0, 1, 2, ...

Standard Deviation of a Poisson Distribution

For a Poisson distribution, the mean (λ) is equal to the variance. Consequently, the standard deviation (σ) is the square root of the mean:

σλ

Steps to Find the MLE of the Mean (λ)

To find the MLE of the mean (λ), given a random sample X1, X2, ..., Xn drawn from a Poisson distribution, follow these steps:

Likelihood Function

The likelihood function L λ for the sample is the product of the individual probabilities. For a sample of X1, X2, ..., Xn, the likelihood function is:

Lλ∏i1nPX_ix_i∏i1nλ^x_ie-x_i!

This can be simplified to:

Lλλ∑i1nx_i∏i1nx_i!e-nλ

Log-Likelihood Function

To simplify the maximization process, we take the natural logarithm of the likelihood function to obtain the log-likelihood. This is done as follows:

logLλ∑i1nlog(λ^x_ie-x_i!)∑i1nx_ilogλ-λ-logx_i!

This can be further simplified to:

logLλ(∑i1nx_i)logλ-nλ-∑i1nlogx_i!

Differentiation of the Log-Likelihood Function

Next, we differentiate the log-likelihood function with respect to λ and set the derivative equal to zero to find the MLE of λ:

?logLλ?λ∑i1nx_iλ-n0

Upon rearrangement, we get:

λ∑i1nx_in

Thus, the MLE of λ is:

λhX_1n∑i1nX_i

MLE of the Standard Deviation

Since the standard deviation (σ) is related to λ by:

σλ

We can find the MLE for the standard deviation as follows:

σhλhX_

Conclusion

In conclusion, the MLE of the standard deviation of a Poisson distribution with an unknown mean, based on a random sample, is:

σhX_1n∑i1nX_i

This method provides a systematic way to estimate the standard deviation using your sample data. If you have any further questions or need clarification on any steps, feel free to ask!