Random variables can be broadly classified into three types:
- random variables whose range is finite,
- random variable whose range is countably infinite and
- 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.
Random variables that can take an uncountably infinite number of values are not discrete
Almost all the probabilistic models used in artificial intelligence contain random variables.
In theory, one 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 the 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.
My doubt is:
Do researchers use random variables of type 2 or type 3 during the 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.
