Probability Theory MOC
Probability theory concerns the axiomatisation of probability models. Traditional models rely on the concepts of Set theory MOC and Measure theory MOC. It is often broken into the two branches of discrete probability and continuous probability.
Foundations
At the foundation of probability theory is the Probability model
By introducing the concept of a Real random variable, we can give defined values to different outcomes and therefore make comparisons between them. For all random variables we may speak of
- Cumulative distribution function
- Expectation
- Standard deviation
- Statistical moment
- Moment-generating function
- Probability-generating function
The Discrete random variable and Continuous random variable are starting points for Discrete probability and Continuous probability respectively.
Discrete probability
Discrete probability concerns the Discrete random variable. Common distributions of such variables include
Continuous probability
Continuous probability concerns the Continuous random variable, and so it uses many of the tools of calculus and analysis. Common distributions include
Higher concepts
- Characteristic function (probability)
- Multivariate random variable
- Random function
- Inequalities in probability MOC
- Convergence concepts in probability MOC
Statistical inference
Using a probability model for any real-world applications involves the assumption that the parameters of a distribution are known. Generally, this involves extrapolating from a Random sample, which may then be used to generate statistics, some of which are estimators — both of which are a form of Real random variable. Importantly, the 𝜇-estimator gives rise to the Central limits theorem.
Another important aspect of statistical inference is the Statistical hypothesis, a guess on how a random variable is distributed.