the concepts of finite and countably infinite are discussed

the concepts of finite and countably infinite are discussed for continuous and discrete probability functions. What is countably finite / infinite?

Solution

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Let us consider an experiment where the outcome depends on chance. So the outcome of the experiment be as X, called as a random variable. The sample space of the experiment is the set of all possible outcomes. If the sample space is either finite or countably infinite, the random variable is said to be discrete.

Countably Infinite / Finite :

For example, if the considered sample space has an infinite number of points/values, then the process a distribution function is defined depends upon whether or not the sample space is countable. A sample space is countably infinite if at all the elements can be counted, that means that they can be put in an one-to-one correspondence with the positive integers, and finite in the second case.

To be noted that a uniform distribution function cannot be defined on a countably infinite sample space that could be acheived on a finite sample space.

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