ABSTRACT
Given a probability space https://www.w3.org/1998/Math/MathML"> ( S , E , μ ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003260547/1539354b-4ff6-4752-881b-983ba2324ebc/content/math4_1.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> and a random variable https://www.w3.org/1998/Math/MathML"> X : S → ℝ , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003260547/1539354b-4ff6-4752-881b-983ba2324ebc/content/math4_2.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> one is often interested (see Chapter 5, for example) in investigating properties of a collection of random variables https://www.w3.org/1998/Math/MathML"> { X j } j = 1 N , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003260547/1539354b-4ff6-4752-881b-983ba2324ebc/content/math4_3.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> where N can be finite or countably infinite, which are independent and identically distributed with X (i.i.d.-X or simply i.i.d.):
Independence is characterized in Definition 3.47.
Identically distributed with X reflects the notions of Definition 3.1, that all X j have the same distribution function as X.
