A Poisson distribution can be thought of as an effective tool for analysis and prediction. Its name comes from the French mathematician, Pierre-Simon Huyseville Poisson (often referred to as PV or Poyer), who first came up with a solution to the problem of how to best fit a probability distribution to a set of real data. His resulting algorithm was eventually patented in 1849. The main idea behind a Poisson distribution is that the distribution of the random variable is characterized by mean and variance components that are independent of each other but strongly dependent upon the values of the other component. As such, the mean and variance components of the distribution will cancel each other out when the data distribution used by the model is plotted on the x-axis and the data point selected as the point curve of the distribution is plotted on a y-axis.
Using the PV formula, PV estimates the probability density of the distribution. The PV estimate is often used as a way of fitting a normal curve to the data set being analyzed. For instance, if a normal exponential curve is plotted on the x-axis, the estimated probability density can be plotted on the y-axis as well. This makes the PV formula especially useful in the analysis of logistic regression and statistical distributions. Another application of the Poisson distribution is in the probability distribution function, which is often used as an adjoint variable in a multivariate statistical analysis where the data set is distributed using a Poisson distribution function.
To derive the PV distribution, the likelihood ratio function, the parameters, and the chi-square distribution are necessary conditions for the data distribution. chi-square distribution functions are often fitted to the data set using finite difference estimates, or F-statistics, as these are based on the arithmetic mean of the observations and provide an estimate of the standard deviation. The chi-square distribution can also be fitted to the normal curve via the method called the Laplace approximation. When fitting a Poisson distribution to a normal curve, the least significant number required by the data set must be allowed to deviate from zero. In the final analysis, the chi-square distribution should be compared to the normal curve to ensure that it has a similar shape and fits the data better.