If X in P(X) is considered a random variable because it varies among the occurrence the particular scenario which it may occur (such rolling a number on a die), can we still call X a “random variable” if it’s technically predictable? (Such as if X were like “getting a face on a die when you roll a die” or “getting a 7 on a die when you roll a 6 sided die.”)

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    $\begingroup$ We can treat any object that can be formally described by a distribution function as being a random variable. This holds even if the distribution function is degenerate. $\endgroup$ – Aaron Hendrickson Feb 27 at 15:54
  • $\begingroup$ See stats.stackexchange.com/questions/50/… $\endgroup$ – kjetil b halvorsen Feb 27 at 16:06
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    $\begingroup$ In the language of the "tickets in a box" metaphor at stats.stackexchange.com/a/54894/919, nothing says you can't write the same value on every ticket. That is, there is no mathematical, logical, or statistical reason to exclude random variables $X:\Omega\to\mathbb{R}$ that are constant functions. $\endgroup$ – whuber Feb 27 at 16:42

Sure, we can do so. Such random variables simply have degenerate densities or probability mass functions, such that the probability of an outcome is 1, or the outcome we are interested in may have a probability of 0.

Such degenerate cases can easily come up, e.g., when we are looking at conditional probabilities. Let's throw a standard six-sided die.

  • What is the distribution of the parity of the side shown conditional on throwing a 1, 2 or 3?
  • What is the distribution of the parity of the side shown conditional on throwing a 2, 4 or 6?

The second case is degenerate: the conditional probability is 0 (for an odd side showing) or 1 (for even).


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