ABSTRACT

An obvious example of the subject of this chapter is the probability distribution of the number-correct score of a test taker on a test with dichotomous or polytomous items. However, similar types of distributions are met in statistical tests of cheating on tests, for instance, as null distributions of the number of matching responses between two test takers suspected of answer copying or the number of answer changes by a single test taker reviewing his answers in the detection of fraudulent erasures on answer sheets. In each of these cases, the desired distribution is found as the convolution of more elementary distributions. Although they may seem to lead to complicated combinatorics, a simple recursive algorithm exists that allows us to evaluate the convolutions efficiently and accurately. The same algorithm can be used to evaluate the posterior odds of certain events given sum scores or the combinatorial expressions that arise in conditional maximum-likelihood (CML) estimation of the parameters in the Rasch model. In addition to these discrete cases, the chapter addresses the distribution of the total time spent by a test taker on a test, which is found as the convolution of the distributions of his response times (RTs) on the items.