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Most functions that generate random numbers are described as:

Returns a pseudorandom, uniformly distributed int value between 0 (inclusive) and the specified value (exclusive)

If I use it to generate a number in the range [0..9], does uniformly distributed mean that the probability for 3 and 8 is the same?

As I understand it, if I run the function a sufficient number of times, I'll have the same number of occurrences for each number, 0 through 9. Is this correct?

Alternatively, would we instead have a normal distribution where we would get more occurrences of the numbers 4, 5, and 6, as they are closer to the mean and hence have a higher probability of occurring?

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Uniform distribution is when all values have the same probability. A random uniformly distributed integer in the range $0,\ldots,9$ will attain each of the values with probability $1/10$. In practice, you function is not truly random but only pseudorandom, so the probabilities won't be exactly $1/10$ but only very close to $1/10$.

Normal distribution is a probability distribution on real numbers. If we "bin" them then we get a normal distribution on the integers. If we "cap" it (there are several ways of doing this) then we get a probability distribution on a finite set of integers.

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  • $\begingroup$ thanks, so all my assertions in the question body are correct, right? Normal distribution is a probability distribution on real numbers. - can you please clarify? $\endgroup$
    – Max Koretskyi
    May 17, 2017 at 15:45
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    $\begingroup$ The normal distribution is a probability distribution (in fact, a family of probability distributions) on the real numbers. Sometimes people say "normal distribution" when they actually mean some discretization of the normal distribution. $\endgroup$
    – Yuval Filmus
    May 17, 2017 at 15:56
  • $\begingroup$ _ If we "bin" them then we get a normal distribution on the integers. If we "cap" it_ - thanks, where can I read about bin and cap you're talking about? $\endgroup$
    – Max Koretskyi
    May 18, 2017 at 16:06
  • $\begingroup$ Some question on this site mentioned discretized normal distribution, but I don't of any other particular source, though I imagine it's common in some circles. $\endgroup$
    – Yuval Filmus
    May 18, 2017 at 19:05

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