> For the complete documentation index, see [llms.txt](https://zavalab.gitbook.io/statschemebook/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://zavalab.gitbook.io/statschemebook/chapter-7-decision-making-under-uncertainty/01-motivation.md).

# Motivation

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**Slides:** [stats\_ch7\_decisionmaking.pdf](https://github.com/vzavala/StatsBook/blob/master/Slides/stats_ch7_decisionmaking.pdf) · [4-up handout](https://github.com/vzavala/StatsBook/blob/master/Slides/handouts/stats_ch7_decisionmaking_4up.pdf) · **Code:** [Python notebooks](https://github.com/vzavala/StatsBook/tree/master/Python) · [MATLAB scripts](https://github.com/vzavala/StatsBook/tree/master/Matlab) · [MATLAB live scripts](https://github.com/vzavala/StatsBook/tree/master/Matlab-Live)
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* We have used *statistical techniques* to model known/unknown aspects of a system.
* We now want to use these capabilities to *make decisions*.
* Accounting for uncertainty is *essential* when making decisions (ensure robustness).
* Making decisions under uncertainty boils down to *comparing* RVs and to *shaping* the probability distribution of RVs.
* Discussion will address following specific questions:
  * What can happen if we ignore uncertainty?
  * How do we measure risk/uncertainty?
  * How can we obtain a decision that anticipates and responds to uncertainty?
  * How can we compare random variables?
  * How can we shape probability distribution of random variables?
* These questions are studied in area of *stochastic optimization*.
* This is a broad area that has applications in engineering, science, and finance.
* Consider system:

  $$
  Y=\varphi(X,u)
  $$
* $$X=(X\_1,...,X\_n)$$ is multivariate input RV and $$Y$$ is univariate output RV.
* System model is $$\varphi$$ and we interpret this as performance function (e.g., cost).
* Realizations of $$X$$ propagate through system to generate realizations of $$Y$$:

  $$
  y\_\omega=\varphi(x\_\omega,u),; \omega \in \mathcal{S}
  $$
* Uncertainty in $$X$$ propagates through system to generate variability in $$Y$$.
* e.g., input can be prices of energy that affect profit of a process.
* Can use decision $$u$$ to protect system against variability of $$X$$.