Probability Distribution Functions

1. Bathtub - These are used to describe the failure probabilty of solid-state electronics. These are described in my book titled "Monte Carlo Risk Analysis and Due Diligence of New Business Ventures." A description of this book can be found by proceeding to the "New Book Info" option of the Navigation Bar on the left-hand side of this web page.
   
   
2. Chemcial Reaction Rates and Yields - These generally have to be derived. However, the published literature, NIST Data Book, and the University of Texas Thermodex (see Links) are good places to start the process.
   
   
3. Flat or Constant - Use a constant value (a flat distribution curve) if there is thought to be little or no variation in the distribution.
   
   
4. Gaussangular Distribution^TM - This distribution is one I have developed and used for many years. It was first published in my book titled "Monte Carlo Risk Analysis and Due Diligence of New Business Ventures." A description of this book can be found by proceeding to the "New Book Info" option of the Navigation Bar on the left-hand side of this web page. It has the ability to closely approximate a true Gaussian, an unsymmetrical Gaussian, triangular, and mesa-type distributions. It is very fast to use in calculations (e.g., less than 2 minutes to perform 50,000 iterations on a 40-variable metric with a rather modest 500 MHz PC). The book has includes CD that has an EXCEL macro that uses this methodology.
   
   
5. Gaussian - A "Gaussian-type" or normal distribution is used for most variables that have been scientifically measured or derived, or represent classifications of plants, animals or humans. Significant disadvantages to the Gaussian distribution function are that it symmetric about its "mean," spans the domain of, and is normalized over, all real numbers (from - infinity to + infinity), and is cumbersome to use in digital computers.
   
   
6. Mass and Energy Flow Distributions - Both the University of Texas and NIST are a good place to look for thermodynamic data. (See Links)
   
   
7. Nuclear Reactions - Since Los Alamos first developed Monte Carlo theory, they are the best place to start looking for nuclear cross-section data. (See Links)
   
   
8. Other "Angular" - Use an "angular" distribution that is derived to be a good fit to the data represented by the more conventional Distribution Functions (Gaussian, etc.) These angular Distribution Functions are constructed by a series of straight-line approximations to the more complex continuous distribution functions. This significantly reduces the computation time required in Monte Carlo calculations while retaining the validity of the simulation. Further, these angular distribution functions allow for the more "realistic" upper and lower bounds that are not plus and minus infinity nor must they be symmetric about the most likely value. The Gaussangular Distribution^TM is described above. This methodology can be applied to fit any type of data to any "realistic" distribution function! These angular Distribution Functions can be created by specifying the absolute minimum value, the most likely value, and the absolute maximum value.
   
   
9. Poisson - Use a "Poisson-type" distribution for the failure of mechanical manufacturing or "process" components, or any rare phenomena. It serves as a useful probability model for events occuring randomly over time or space when all that is known is the average number of occurences per unit time or space. The only input required is the average rate of occurences per unit time or space. The "variance" is always equal to the "mean" of a Poisson Distribution.
   
   
10. Triangular - Use a triangular distribution if you know very little about the distribution. The Triangular Distribution Function can be completely defined by knowing the absolute minimum value, the most likely value, and the absolute maximum value. It can also be easily skewed to match reality.