Discrete deconvolution

This dissertation studies the deconvolution of the ternary (three-outcome) random variable into two Bernoulli random variables and the development of a statistical test to see if the probability of “success” in the two Bernoulli random variables is statistically significant. A Bernoulli random variable has only two possible values. One of the values is termed a “success” and the other value a “failure.” If you assign a 1 to the “success” outcome and 0 to the “failure” outcome, the result of the sum of two Bernoulli random variables is a ternary random variable with values of 0, 1, and 2. The problem dealt with in this dissertation is that if you observe the ternary random variable that is the sum of two Bernoulli random variables, can you derive the distribution of the two Bernoulli random variables distributions of probability and test to see if their success probabilities are different using sample observations derived from an experiment where the ternary distribution is observed? Chapter One discusses the motivation, description, possible applications, and similar problems in the literature.; In order to understand deconvolution, the concept of convolution, the mathematical opposite of deconvolution, is discussed in Chapter Two as a way of introducing deconvolution. There is a direct connection between finding the probabilities of a convolution when the generating distributions are discrete. This theory is discussed, and then deconvolution is considered along with the role of independence in the deconvolution problem.; The maximum likelihood estimators of the probability of “success” in the Bernoulli random variables are derived in Chapter Three and used in Chapter Four to develop a test statistic to see if there is a significant difference between the two “success” probabilities.; The test statistic developed in Chapter Four requires that the two Bernoulli random variables that generated the ternary distribution be independent. An analysis deriving the test of independence is completed in Chapter Five.; An example of the analysis necessary to work with data is described in Chapter Six for scientists wishing to use the test statistics developed in this dissertation.