Probability theory and statistics

See also: Introduction to risk analysis, Vose Probability Calculation, Analyzing and using data introduction, Distributions in ModelRisk

This set of topics contains the essentials of probability theory that you need to know as a risk analyst. The topics are split into three groups: the basics, parameters and sample statistical measures, and stochastic processes.

It can be somewhat confronting for the trained statistician to start learning about risk analysis, as the focus is on different things. Some subjects that traditionally get high attention in statistics, are of less importance in risk analysis modeling (and vice versa). Probably this is because most statistics courses focus on applications in data analysis, and less on building stochastic models.

Distributions are a prime example of this: when doing mathematical modeling one quickly learns, that the Normal distribution (which receives a disproportional amount of attention in most statistics courses) is just that: a distribution, one of the many distributions available for building mathematical models.

Of central importance for risk analysis modeling are the stochastic processes. Gaining a good understanding of these will allow you to translate real-world problems to the correct mathematical model.

Another important set of topics about using statistics for presenting results (like MC simulation output results) can be found under Presenting results.

The basics of probability theory

See also: The basics of probability theory introduction

We define some basic statistics in common use. We look at a few probability concepts that are essential to understand if one is to be assured of producing logical models.

Note that this section is designed to offer a reference of statistical and probability concepts. The application of these principles is left to the appropriate section elsewhere in this help file.

• The section on probability equations explains the equations that define probability distributions: pmf, pdf, cdf.

• The section on probability parameters explains the meaning of standard statistics like mean and variance within the context of probability distributions. That comparison of the meaning of these statistics for uncertainty and frequency distributions is discussed elsewhere.

• The section on probability rules and diagrams explains visual ways to depict probability ideas, and rules for the manipulation of probabilities in calculations.

• The section on probability theorems explains some fundamental probability theorems most often used in modeling risk, and some other mathematical concepts that help us manipulate and explore probabilistic problems.

Parameters and sample statistical measures

See also: Parameters and sample statistics introduction

There are five different circumstances in which we use descriptive parameters to describe distributions:

• Frequency distributions of populations

• Frequency distributions of samples

• Probability distributions of random variables

• Uncertainty distributions of real-world parameters

• Frequency distributions of Monte Carlo simulation results

The calculation and interpretation of the statistical measures will depend on which of these five distributions you are describing.

Stochastic processes

See also: Stochastic processes introduction

Stochastic processes are the building blocks of probability modeling and statistics. That means you really need to know them!

We describe the following processes:

• Binomial process

• Poisson process

• Hypergeometric process

• Central Limit Theorem

• Renewal processes

• Mixture processes

There are a number of other stochastic processes, essentially variations of the above, that are used in modeling time series.

Read on: The basics of probability theory