Why should random numbers be drawn from a distribution?

What does it mean to draw a number at random from a normal distribution? Why can't you just "draw" numbers at random? Why do you have to draw random numbers from a distribution?

• I think you should thin about this question in reverse. If you draw random numbers (say for example pulling numbers out of a hat) will it follow a distribution? The answer is yes, although you may not know what the distribution is. So to answer your second questions yes you can just "draw" numbers at random.
– user25658
Commented Sep 23, 2013 at 14:54
• @BabakP Your advice is good, but beware! Not all procedures for creating numbers arbitrarily will have probability distributions. Because this is a subtle concept, here's an explicit example couched in the language of discrete stochastic processes: draw values independently from a Bernoulli$(3/4)$ distribution at times $t\ge 1$ when the integer part of $\log_2(t)$ is even and otherwise draw the values from a Bernoulli$(1/4)$ distribution. What could "the" distribution of this process possibly be? In light of this, one has to question whether your answer to Q2 even makes sense.
– whuber
Commented Sep 23, 2013 at 15:14
• @Peter Interesting thought; but no: make a plot of the expected mean as a function of $t$ to see what's going on. (Use a logarithmic scale for $t$.) There are deep conceptual issues involved here: randomness is not arbitrariness, nor is all randomness describable with a (fixed) probability distribution.
– whuber
Commented Sep 23, 2013 at 16:14
• Can you explain what you mean by "draw numbers at random?", without implying an underlying distribution? I must admit to being skeptical that this could even be a coherent notion. Commented Sep 24, 2013 at 8:02
• Phil, finding numbers in your head is demonstrably not random. Randomness implies certain properties that subjective generation of numbers does not have.
– whuber
Commented Sep 24, 2013 at 14:43