## Esp. Probabilists, Applied Mathematicians, Statisticians

### HPD: The Capability for "SMAO" & Consequent Implications

### 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 (

**RV**s) & Stochastic Variables (

**SV**s)

**, in essence, enabling "Transcendence from the Deterministic to the Stochastic Realm. " This is all due to HPD's**

*with ease***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.

**SA**can handle

**of relationship**

*any type***z**= g(

**X**), for which the distribution for any

**x**

_{i}(of

**X**≡ {

**x**

_{i}}) can be of any type! That it can compute "exact" output distributions means an

**can be used as an**

*output***distribution for the**

*input**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 (**SIVP**s). : 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__On Statistics__**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

: 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__On Mathematics__**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.): 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**On Applied Mathematics****unifying**Applied Mathematics.