Chi-square goodness-of-fit test for the Poisson, binomial and negative binomial distributions

Some tests of chi-square-type are constructed for testing Poisson, binomial or negative binomial hypothesis. Some grouping procedures are discussed when unknown nuisance parameters are estimated from ungrouped data. For alternatives of the above types, some special tests are proposed.
As an example, the data due to Rutherford and his assistants concerning the emission of $\alpha$-particles are considered. These data lead to rejection of the Poisson hypothesis in favour of the binomial alternative. However this does not contradict the exponential model of emission and might be explained by means of «resolving time».