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The Poisson process is a stochastic process, which describes events that occur independently from one another and continuously. Stochastic processes are part of probability theory and are used to describe random processes. They are based on a level of indeterminacy, which means that the final outcome is unknown although some paths and outcomes are more likely than others. (Doob, 1953)
The Stochastic Poisson model has been used to describe processes like rainfall, the telephone calls that arrive at a switchboard, radioactive decay of atoms, and the page views of a website. Its use to describe the decision making of a juror is a relatively new and exotic application.
When applied to decision making a Poisson process can be catalogued as special case of renewal theory (Cox, 1962). This model was proposed by Thomas and Hogue (1976) as a descriptive model in juror decision making.
The model describes the jurors choice making as a two step process. In the first place the juror must consider the evidence to create a final estimate of the weight of the case against or for the defendant. Secondly, each juror has an individual decision criterion that allocates the apparent weight of evidence into "for" and "against" decision zones. According to this model a juror will only decide against a defendant if the weight of evidence exceeds his personal decision criterion. This means that the confidence of a juror in any particular decision will be a mathematical function of the distance between the apparent weight of the evidence and their personal decision criterion. The further away these parameters are the stronger the confidence of the juror in a particular decision.
The indeterminacy of this process lies in the assumption that the apparent weight of evidence, or the perception of the weight of evidence, is randomly distributed among jurors. Thomas and Hogues (1976) decided to use an exponential probability density function to describe the way jurors arrived at a
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