Poisson distribution

Assume that each data point has an error that is independently random and distributed as a Poisson distribution. The log likelihood function, `L(p)`, as a function of the fit parameters, `p`, is minimized using a Gauss-Newton method. Since logarithms are involved, a good first approximation is required before starting the Poisson fit, so try a normal fit first, and use the resultant parameter values to start off the Poisson fit.

Weights do not have meaning, and so are not used, in a Poisson fit.

Assume that each data point, `yk`, has an error that is independently random and distributed as a Poisson distribution, that is,

We want to minimize:

but `∑ln(yk!)` is a constant. So, the goal is to minimize

Consider the Taylor expansion of :

Define:

Then:

Linearize, and the problem reduces to solving the matrix equation

Chi-square of the fit

Normal distribution