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If I have an array {0.1, 0.2, 0.3, 0.4, 0.5}, is this a discrete array? Are the values in it discrete values? (I have this doubt because many materials show that discrete values are usually integers)

Like:if one divided dice-face values by 10, is the sample space of possible values {0.1, 0.2, 0.3, 0.4, 0.5, 0.6} still considered discrete?

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    $\begingroup$ I think you are using the word "variable" incorrectly. A "random variable" is a something like a 5-sided dice roll. Your question seems to be: if one divided dice-face values by 10, is the sample space of possible values {0.1, 0.2, 0.3, 0.4, 0.5} still considered discrete? $\endgroup$
    – AdamO
    Commented May 29 at 16:51
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    $\begingroup$ Why wouldn't the space still be discrete just because you divide by ten? $\endgroup$
    – Dave
    Commented May 29 at 17:03
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    $\begingroup$ @AdamO Die singular, dice plural. (and +1) :) Also: Cathy, not only yes to your question, but a discrete variable can also take infinite numbers like $.\bar{3}$ and transcendental numbers like $\pi$ as values in their probability mass functions. :) Heck, I suppose they could even take p-adic numbers as values. $\endgroup$
    – Alexis
    Commented May 29 at 18:39
  • $\begingroup$ Quick note that in the "real world", you could consider all your variables to be discrete. Because of the limited resolution of your measuring devices. Does that matter practically?. No, it really does not change how you handle these (discrete) variables, which tests you will use, etc... $\endgroup$
    – jginestet
    Commented May 30 at 17:06

1 Answer 1

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Can a finite decimal number be a discrete variable?

This is an abuse of terminology, but the fact is that discrete random variables are not limited to integer or categorical outcomes.

Any probability distribution on that $S = \{0.1, 0.2, 0.3, 0.4, 0.5\}$ sample space will have a certain amount of probability mass on each value. The distribution would be defined by:

$$ P(X = 0.1) = p_1 \ge 0\\ P(X = 0.2) = p_2 \ge 0\\ P(X = 0.3) = p_3 \ge 0\\ P(X = 0.4) = p_4 \ge 0\\ P(X = 0.5) = p_5 \ge 0\\ P(X = 0.6) = p_6 \ge 0\\ \sum_{i = 1}^6 p_i = 1 $$

That is the probability mass function of a discrete random variable.

You can extend this to other finite or even some infinite sets (e.g., Poisson distribution defined on the non-negative integers) to get many examples of discrete random variables that take non-integer values. For instance, what proportion of the pizza do you eat when it is cut in eight slices? That has a sample space of $\{0, \frac{1}{8}, \frac{1}{4}, \frac{3}{8}, \frac{1}{2}, \frac{5}{8}, \frac{3}{4}, \frac{7}{8}, 1\}$.

Here is a fun example for an infinite sample space.

$$ S = \left\{ 1, 2, 3,\dots \right\}\\ X\sim F_X\\ F_X(x) = P(X\le x) = \overset{\lfloor x\rfloor}{\underset{k = 1}{\sum}}\left( \dfrac{1}{2^k} \right) $$

I have used a geometric (ish) series to guarantee convergence to one.

If you want to introduce some irrational numbers, here is an example.

$$ S = \left\{ \pi, 2\pi, 3\pi,\dots \right\}\\ X\sim F_X\\ F_X(x) = P(X\le x) = \overset{\lfloor x/\pi\rfloor}{\underset{k = 1}{\sum}}\left( \dfrac{1}{2^k} \right) $$

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