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1) I am aware that a continuous random variable cannot be obtained from a sample space that is countably infinite or finite. In other words, the sample space of an experiment has to be uncountably infinite in order for one to be able to assign a meaningful continuous random variable. But can one define a discrete random variable on an uncountably infinite sample space? Or does an uncountably infinite sample space only allow us to define a continuous random variable?

2) My second question is somewhat different and relates to continuous random variables specifically. Given a continuous random variable $X$ and a real-valued function $g$, $g(X)$ is a random variable. However, unlike the discrete case wherein functions of discrete random variables are inevitably discrete, this need not be the case for functions of continuous random variables. If $X$ is continuous, $g(X)$ can be continuous or discrete. I find this confusing because in order for us to define a continuous random variable $X$ to start with we need an uncountably infinite sample space. But given that uncountably infinite sample space how can one associate with it a discrete random variable $g(X)$? Finally, does one define a probability mass function for $g(X)$ if it turns out to be discrete?

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Answering your first question, if your sample space is uncountable, or, you can not have one to one mapping among the set of natural numbers and the elements of the set, then how will you discretise? Or in other words does the elements of the sets are discreet signify each elements of the set can be mapped to an unique natural number?

Answering your second question, let your function g be a discretiser . In this case, although the your input sequence of random variables can be continuous and the discretised sequence be random variables as well, because your discretised sequence can be mapped to the set of natural number on a one to one basis, your output sequence will not be continuous any more.

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  • $\begingroup$ But then $g(X)$, the discrete random variable, would be associated with an uncountably infinite sample space, which you said is not possible. How do you explain that? $\endgroup$ – 0MW Jul 15 '18 at 12:09
  • $\begingroup$ I’m also wondering, given an uncountably infinite sample space, if we define the degenerate random variable $X$ that takes a constant value (say 2) for all outcomes, is this a discrete or continuous random variable? $\endgroup$ – 0MW Jul 15 '18 at 12:19
  • $\begingroup$ If g(X) has only a discrete set of possibilities, it is discrete by definition. It doesn't matter whether or not X is discrete. $\endgroup$ – herb steinberg Jul 15 '18 at 15:10
  • $\begingroup$ You only have to worry about the range of the function g and by definition g has countable many elements. Note that when you talk about random variables you basically talk about a sequence. If by definition your function has Singleton range space, do you think that your sequence still represents a random sequence? $\endgroup$ – Eval Jul 15 '18 at 15:21

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