*Implications*

*Implications*

**On Fundamental and Future Thinking due to Being Able to Compute Resulting Distributions: **

**Distributions are liberated (thus is Applied Probability):**The currently known**types**(~ 93) of continuous distributions (except for a few very simple ones such as Uniform & Exponential) are but*mathematical transformations*from the Normal; they are just a**micro**set of**discrete**points in the*infinite-dimensional space*of distributions!! But, resulting distributions depend on causal*relationships*which can be*anything*!

Thus, for example: For Design Engineering, the distribution of*any level*of parameter,**y**,*cannot be Lognormal*since the hierarchical relationships between the lowest level Parameter and**y***do**not**logistically compound to***a lognormal function**!!!

**For futurists,****perhaps we’ll all be able to think stochastically?:**For any**z**= g(**X**), since computing**z**is the*same as computing its distribution*, we could think of computing stochastically as a*transcendence*from the Deterministic Realm! This could mean we have**historically**been**thinking limitedly**, but**will have the capacity**to*think stochastically*in the future!!

**On Design Engineering (DE) of Complex Electro-Mechanical Systems **

- HPD enable practitioners to
**properly**handle Variability and reach Design decisions. Since addressing Variability comprises**most**of DE’s work content in many manufacturing industries (especially of complex Electro-mechanical products), this enables**significantly****reducing**(perhaps**by 2/3**!!)**resources, development time, and # of hardware & prototype builds**! - Our advancements in the following significantly contribute to
**enlightening fundamental concepts and enhancing DE practices**:- Rigorizes Latitude; clarifies distinction between Latitude & Robustness
- Defines C
_{pk}s at any level;**enables Multi-Level C**(e.g., parts to system) (_{pk}propagation*Management asked for this*!) - Clarifies Analysis, Synthesis, & Optimization for handling simple to complex problems
- Finds Stochastic Operating Windows (º Feasible Design Regions) and the most Robust points
*within*them, comprehending all internal & external noises and performance targets

- As importantly, engineers would
**understand SO much more**technical info about their systems (local or global) with HPD definitions, work processes, and results!

**For CAE, PLM, MBSE Systems and Digital Twins Initiatives**

- Since Designs must specify Nominals

__and__Tolerancesfor every part feature

__and__operating setpoint (= some of the critical parameters),

**both**Nominals & Tolerances must be carried in those systems. Likewise, a Digital Twin must consist of

**both**such values.

**On All Other Application Areas where Variability Needs to Be Addressed **

- For areas such as Material Science, Chemical Engineering, parts manufacturing industries, and
**ALL**technical fields (except computer science) the same HPD Software & Methodology capabilities apply as for DE. Importantly, HPD makes it all**easier**,**correct**, and**comprehensive**. - For Operations Research Needs, terminologies differ from DE’s, but examples such as Risk Analysis & Stochastic Optimization for simple to complex situations are handled just as for DE
- For Miscellaneous Other Areas: Class projects conducted by Univ. of Rochester
**students**who took**HPD-based courses**covered a wide range of problem types of**their own choosing**. Examples included many that relate to their Master’s theses in other disciplines (e.g., “Arsenic in Bangladesh’s Groundwater” or “Air Induction Rate of an Anti-Swirl Stator Mixing Tank Design”) and some to their interests (e.g., to “Variability in Tuba Playing” and “Quality of Calculus Teaching”). All were complex and required FoVs (Flow of Variabilities) as the*first*step of their structuring a total Stochastic Model.

**On Mathematics, especially Probability, Applied Mathematics, & Statistics **

**For specifics, see ****HPD/Description C****. Here are some major summary points:**

- For
**Probability**, liberated the concept of “Types” of distributions and proved inadequacy of certain assumptions and many other concepts - Liberated
**Applied Probability**thus enabling Transcendence to the Stochastic Realm (think & work naturally with**RV**s &**SV**s) - For
**Mathematics in general**, enabled the concept that Deterministic Realm is but a**sub-Realm**of the Stochastic Realm - Enabled unifying
**Applied Mathematics**by providing a comprehensive capability for**SMAO**(Stochastic “Modeling,” Analysis, & Optimization) and addressing when are Deterministics & Stochastics appropriate.

**On Course Contents in Academia**

Revolutionizing Stochastics will certainly alter how we teach **Probability, Applied Mathematics** (see 4th bullet above), **Statistics** and **Engineering**. It will also imply many needed additions regarding Stochastic treatments in other disciplines where Variability is important (e.g., Business, the Sciences, . . .)