Estimation of Parameters and Fitting of Probability Distributions

In this chapter, we discuss fitting probability laws to data. Many families of probability laws depend on a small number of parameters; for example, the Poisson family depends on the parameter λ (the mean number of counts), and the Gaussian family depends on two parameters, μ and σ . Unless the values of parameters are known in advance, they must be estimated from data in order to fit the probability law. After parameter values have been chosen, the model should be compared to the actual data to see if the fit is reasonable; Chapter 9 is concerned with measures and tests of goodness of fit. In order to introduce and illustrate some of the ideas and to provide a concrete basis for later theoretical discussions, we will first consider a classical example—the fitting of a Poisson distribution to radioactive decay. The concepts introduced in this example will be elaborated in this and the next chapter.