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Modeling and Analysis of Compositional Data by Vera Pawlowsky-Glahn, Juan Jose Egozcue, Raimon Tolosana-Delgado

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Appendix B

Random variables

In a general sense, random variables are measurable functions from a probability space on a sample space. This sentence involves some mathematical concepts that are recalled here for easy reference when reading Chapter 6. Some are elementary, but others are linked to abstract measure theory. Most concepts can be easily found in textbooks on probability, for instance, Mood et al. (1986), Ash (1972), or Feller (1966). The first section of this appendix presents the definitions and first properties of probability spaces and random variables. The second section is a reference of the standard description of a probability measure using the cumulative distribution function and the probability density.

B.1 Probability spaces and random variables

A probability space is made of three mathematical objects, an arbitrary set, here denoted by b03-math-0001, a b03-math-0002-field defined on b03-math-0003, and a probability measure. The set b03-math-0004 has finite or infinite cardinal, that is, the number of its elements is finite or infinite.

To define a probability space, a -field must be defined in .

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