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Random variables can be classified into three types:

  1. random variables whose range is finite,
  2. random variable whose range is countably infinite and
  3. random variables whose range is uncountable.

Random variable is called discrete if its range (the set of values that it can take) is finite or at most countably infinite.

A random variable that can take an uncountably infinite number of values is not discrete

Almost all the probabilistic models used in artificial intelligence contains random variables.

In theory, once can deal with all three types of random variables. For suppose, in reinforcement learning or probabilistic graphical models, we can take any type of random variables as state or action spaces (in RL) and as nodes (in PGM) and can analyze.

But, in several textbooks, most of the analysis is restricted to random variables of first type. The reason they mention is "to make analysis easy". It will be complex if we deal with either type 2 or type 3 random variables. So, textbooks and materials generally prefer analysis with type 1 only for.

My doubt is:

Do researchers use random variables of type 2 or type 3 during implementation of (any) AI tasks? Is it impossible to use them due to their (infinite) cardinality? If possible, please provide an example mechanism for implementing such random variables.

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