In data analysis, especially the number of data is small, in order to found out the parameter of the distribution, which fit the data the best, maximum likelihood method is a mathematical tool to do so.

The ideal can be found in Wikipedia. For illustration, I generate 100 data points from a Gaussian distribution with mean = 1, and sigma = 2.

In Mathematica,

Data = RandomVariate[NormalDistribution[1, 2], 100]

MaxLikeliHood = Flatten[Table[{
mean,
sigma,
Log[Product[PDF[NormalDistribution[mean, sigma], Data[[i]]] // N, {i, 1,100}]],
},
{mean, -3, 3, 1}, {sigma, 0.5, 3.5, 0.5}], 1]

This calculate the a table of mean form -3 to 3, step 1, sigma from 0.5 to 3.5 step 0.5. To find the maximum of the LogProduct in the table:

maxL=MaxLikeliHood[[1 ;; -1, 3]];
Max[%]
maxN = Position[maxL[[1 ;; -1, 3]], %]
maxL[[Flatten[maxN]]]

The result is

{{1,2.,-217.444}}

which is the correct mean and sigma.

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