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Since the probability of a continuous variable $X$ assume any particular value is always zero, how can any sample be obtained from such distribution? How can we obtain, e.g., the set of observations $\{0.3, 1.1, -0.5, -0.6, 0.9\}$ from a normal distribution, when $P(X = 0.3) = P(X = 1.1) = \ldots = 0$?

Is this happening because of measurement errors? Like there will never be any human weighing exactly 200 lbs, a bottle with exactly 1L of water, and no software capable of generate any number with infinite decimals.

Lenora
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    One solution--not really a very good one theoretically or computationally, but an insightful one--is to view all such data as implicitly reflecting interval measurements. Standard engineering rules of thumb, for instance, state that a value reported as "$0.3$" with one decimal place ought to be considered as the interval from $0.25$ to $0.35.$ The correct probability to ascribe to "$0.3$" therefore is the chance that $X$ lies somewhere in this interval: and that chance is always nonzero when $X$ has any (nondegenerate) Normal distribution. – whuber May 24 '22 at 22:23
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    Another solution, inspired by your title, is to divide and conquer. To draw a sample from any distribution start by flipping a coin. If it's heads, focus on the upper half of the distribution; otherwise, focus on the lower half. In either case you are now working with a truncated distribution, but it's exactly the same kind of object. So repeat: keep flipping coins and choosing halves. Very soon your truncated distribution will be indistinguishable from a constant value, so declare the process finished and use that value. – whuber May 24 '22 at 22:34
  • Thank you very much, @whuber. That was precisely the kind of answer I was needing. The word 'truncated was in my mind when I was writing my question.

    By the way, I don't know if it was you, but my question was assigned as a duplicate. I read the related question and all the answers, but I do not agree on this linking. My question was more 'technical' than theoretical: precisely on how the sample could exist, just as you answered.

     – Lenora May 24 '22 at 23:27
  • I did not close your question as a duplicate, Lenora. When I encountered it, it had already been closed. Feeling that you might be looking for some expansion on the duplicate posts, I offered my comments. – whuber May 25 '22 at 12:41
  • Thank you once again. – Lenora May 25 '22 at 16:39

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