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