# Description C – For Mathematicians

## Esp. Probabilists, Applied Mathematicians, Statisticians

### What Does SMAO Stand for?

SMAO (or SM, SA, & SO) ≡ Stochastic Modeling, Analysis, & Optimization. This is in contrast to Deterministic MAO which most in the technical community are familiar with.  Since attacking real-world problems mostly requires Stochastics, SMAO is needed. By providing extremely user-friendly software suites for SA** & SO, and a novel methodology for SM, HPD enables one to think & work naturally with Random Variables (RVs) & Stochastic Variables (SVs) with ease, in essence, enabling "Transcendence from the Deterministic to the Stochastic Realm. " This is all due to HPD's revolutionizing how we model, analyze, & optimize. Note: How to structure a problem and Model it Stochastically, taking into account all variables whose Variability must be addressed, is as important as having the SA & SO software.

** HPD's SA can handle any type of relationship z = g(X), for which the distribution for any xi (of X ≡ {xi}) can be of any type! That it can compute "exact" output distributions means an output can be used as an input distribution for the next level of, or any other, SA (just as in Deterministics), thus enabling conducting SA for a Flow of Models (often needed for addressing a complex problem) to yield a Flow of Distributions (which has especially important implications on application to Systems).

### Important Implications on Probability

• The capability described in ** above means that HPD has liberated the concept of "Types" of distributions since now there is no limit on Types (i.e., there is an infinitude of Types). Thus the ~93 currently "named" types of continuous distribution types are just discreet points in the (infinite) space of distribution types.
• Comparing results from the ** capability to another tool in the HPD's SA suite has uncovered the inadequacy in distribution-fitting with Pearson Systems, thus in previously assumed adequacy of fitting with 4 moments.
• Clarifies the difference between an RV & SV (the latter is needed for Stochastic Optimization & Stochastic Processes)
• HPD_VA enables significantly simplifying the understanding & teaching of Probability (HPD way ≡ the 21st century way).

### Important Implications on Applied Probability(including Statistics)

• Liberates Applied Probability (there are exclusions, e.g., Bayesian probability - which instead can be handled with our modeling technique).
• HPD's Sensitivity/Contribution Analysis, based on rigorous mathematics, far supersedes existing techniques (which have gross omissions/errors), thus can correctly reduce the size of Stochastic problems.
• HPD's Stochastic Optimization significantly supersedes existing techniques (that for Design Engineering has severe limitations and a fundamental fallacy, and that for Operations Research is in nascent stage)
• HPD’s Methodology shows how to apply one of its tools to Stochastic Initial Value Problems (SIVPs).
• On Statistics: Renders Hypothesis Testing meaningless since a distribution can be of any type.
• On Statistics: One of HPD's "Miscellaneous"; tools (from our HPD_M suite) fits data/moments to the complete set of Pearson System distribution types (i.e., it does not default to a Normal distribution fit for some situations as existing tools do).

### Important Implications on Mathematics/Applied Mathematics

• On Mathematics:  HPD enables rigorously considering the Deterministic Realm as a sub-realm of the Stochastic Realm. Let {DM} & {SM} stand for the classes of Deterministic & Stochastic Models, respectively. One may consider that {DM} ⊂ {SM} since a DM is a degenerate SM with delta functions for distributions of the model's variables. Thus HPD enables perceiving Stochastics as unifying Probabilistics & Deterministics; a liberating concept, indeed. (Note: We use Realm because it is much "larger" than the concept of "Space" which was adequate for Deterministics, e.g., in the "space" of Reals.)
• On Applied Mathematics: In the same sense as above, but more specifically pertaining to (a) HPD’s capabilities, (b) at what point in a product development process (or decision-process) are Deterministics & Stochastics more appropriately needed, and (c) handling both the Forward & the Inverse problems, HPD enables unifying Applied Mathematics.