# Binned Log-Maximum Likelihood

I have a dataset describing, say, radioactive decays. I studied the binned maximum likelihood method for estimating the mean lifetime. Since I am new to Mathematica, I am trying to understand how to estimate the mean lifetime $$\tau$$ using the binned maximum likelihood method. I wonder if Mathematica has algorithms that find the estimate using the binned maximum likelihood method, or if I should design the algorithm from scratch. So far, I have the log-likelihood function:

$$\sum_{i=1}^{N}n_{i}\log (e^{-t_{i}/\tau}-e^{-(t_{i}+\delta)/\tau})$$

where $$n_{i}$$ is the number of detections in the $$i$$th bin. The bins correspond to the line segments $$[t_{i},t_{i}+\delta]$$ where $$\delta$$ is the bin width.

I am not sure how to find the maximum of this function using Mathematica.

Furthermore, I am facing the problem of how to calculate $$n_{i}$$ given the raw data, which is a list of lifetimes in no particular order.

• Can you provide more info? The form of the data in the dataset, some background on the method you want to use, ... – MarcoB Mar 4 at 3:37