Bayesian Networks, 2nd Edition

B

Probability Distributions

B.1 General Features

A probability distribution is a function that assigns a probability to each measurable subset of a set of events. It is associated with a random variable, here denoted as X.

A discrete probability distribution can assume only a finite or countably infinite number of values U, such as

$$\begin{array}{lllll}\{A,B,C,D\}\hfill & \text{or}\hfill & \{[0,5],(5,7],(8,10]\}\hfill & \text{or}\hfill & n\in \mathbb{N},\hfill \end{array}$$(B.1)

and is characterised by a probability function $\mathrm{Pr}(·)$ that satisfies

$$\begin{array}{llll}{\displaystyle \sum _{u\in U}\mathrm{Pr}}(X=u)=1\hfill & \text{and}\hfill & \mathrm{Pr}(X=u)\in [0,1]\hfill & \text{for}\text{all}u.\hfill \end{array}$$(B.2)

A continuous probability distribution must assume an infinite number of values U, typically ℝ or an interval of real numbers, and is characterised by a density function $\mathrm{f}\mathit{}\mathrm{(}\mathrm{\cdot }\mathrm{)}$ that satisfies

$$\begin{array}{llll}{\displaystyle {\int }_U\text{f}}(X=u)du=1\hfill & \text{and}\hfill & \text{f}(X=u)\mathrm{⩾}0\hfill & \text{for all }u.\hfill \end{array}$$(B.3)

$\mathrm{Pr}(u)$ and $\mathrm{f}(u)$ are often ...