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Articles 1 - 4 of 4
Full-Text Articles in Probability
Probability Models With Discrete And Continuous Parts, James E. Marengo, David L. Farnsworth
Probability Models With Discrete And Continuous Parts, James E. Marengo, David L. Farnsworth
Articles
In mathematical statistics courses, students learn that the quadratic function E ((X – x )-squared) is minimized when x is the mean of the random variable X, and that the graphs of this function for any two distributions of X are simply translates of each other. We focus on the problem of minimizing the function defined by y ( x) = E ( IX – xI-squared ) in the context of mixtures of probability distributions of the discrete, absolutely continuous, and singular continuous types. This problem is important, for example, in Bayesian statistics, when one attempts to compute the decision …
Probability Models And Compounding, David L. Farnsworth, James E. Marengo
Probability Models And Compounding, David L. Farnsworth, James E. Marengo
Articles
We present the case that the ideas contained in a particular sequence of formulas are important in probability and statistics. The synthesis offered by the concepts in the sequence can be very valuable. Facility with this sequence and its underpinnings should be in the skill set of anyone who uses or studies probability or statistics. For illustrative purposes, we give applications to mixture distributions and Bayesian analyses.
A Geometric Derivation Of The Irwin-Hall Distribution, James E. Marengo, Lucas Stefanic, David L. Farnsworth
A Geometric Derivation Of The Irwin-Hall Distribution, James E. Marengo, Lucas Stefanic, David L. Farnsworth
Articles
The Irwin-Hall distribution is the distribution of the sum of a finite number of independent identically distributed uniform random variables on the unit interval. Many applications arise since round-off errors have a transformed Irwin-Hall distribution and the distribution supplies spline approximations to normal distributions. We review some of the distribution’s history. The present derivation is very transparent, since it is geometric and explicitly uses the inclusion-exclusion principle. In certain special cases, the derivation can be extended to linear combinations of independent uniform random variables on other intervals of finite length.The derivation adds to the literature about methodologies for finding distributions …
The Conditional Poisson Process And The Erlang And Negative Binomial Distributions, Anurag Agarwal, Peter Bajorski, David L. Farnsworth, James E. Marengo, Wei Qian
The Conditional Poisson Process And The Erlang And Negative Binomial Distributions, Anurag Agarwal, Peter Bajorski, David L. Farnsworth, James E. Marengo, Wei Qian
Articles
It is a well known fact that for the hierarchical model of a Poisson random variable Y whose mean has an Erlang distribution, the unconditional distribution of Y is negative binomial. However, the proofs in the literature provide no intuitive understanding as to why this result should be true. It is the purpose of this manuscript to give a new proof of this result which provides such an understanding. The memoryless property of the exponential distribution allows one to conclude that the events in two independent Poisson processes may be regarded as Bernoulli trials, and this fact is used to …