As a first application of the maximum-likelihood method, we consider the example of the measurement of a physical parameter _{0}, where x is the result of a particular type of measurement that is known to have a measuring error . Then if x is Gaussian-distributed, the distribution function is
For a set of N measurements x_{i}, each with its own measurement error _{i} the likelihood function is
then
(5) |
The maximum-likelihood solution is
(6) |
Note that the measurements must be weighted according to the inverse squares of their errors. When all the measuring errors are the same we have
Next we consider the accuracy of this determination.