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I had a discussion with a professor, who came with this thought: When you randomly sample from a Normal distribution of fixed mean and variance, how can you claim that the sample is random when actually the samples are from a fixed distribution? Are those samples really random? or "random"? I will appreciate if this line of thought can be discussed further. I happen to somewhat agree with his thoughts, am I or he missing some fundamental things here?

PS: When you do this operation on a computer you call it pseudo-random. Assume that a person sampled the samples instead of a computer.

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  • They're random but not all numbers have an equal probability of coming up. Numbers closer to the mean are more likely to come up. It's not a "uniform" distribution.
    – user935
    Dec 9, 2017 at 0:07
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    "When you randomly sample from a Normal distribution of fixed mean and variance, how can you claim that the sample is random" <- this is too easy to answer; I don't get it. Given you randomly sample, your sample is obviously random (if it weren't, how is it you can say you randomly sampled?). It's just tautological. What is your question?
    – H Walters
    Dec 9, 2017 at 11:25
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    Also: "When you do this operation on a computer you call it pseudo-random." ...this only follows if your computer uses a pseudo-random number generator. Computers have RNG's that aren't PRNG's (which works by collecting "entropy" from the environment using inputs). But ignoring that, sure, computers are PRNG's... are you analogously asking then if people are good sources of randomness maybe?
    – H Walters
    Dec 9, 2017 at 11:30
  • I've read that people are bad sources for generating random results when they're told to do something randomly.
    – jjack
    Dec 9, 2017 at 13:37
  • @barrycarter a normal random variable can be constructed using uniform random variable.
    – Ruthvik Vaila
    Dec 11, 2017 at 19:08

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You're conflating a random sample with some measured characteristic of that sample. Suppose you're studying peoples' heights. And suppose you have some procedure that really, really selects a random sample of people from the total population. Then, despite your people sample being really random, the collection of numbers representing their measured heights are likely normally (or thereabouts) distributed. The randomly selected people themselves have no "distribution", per se. Only the measured observable, height.

So suppose instead you had a hat full of one million slips of paper, numbered (you guessed it) one to a million. Then, if you truly randomly select a sample of slips from the hat, you'd indeed expect a uniform distribution of numbers. If you found a normal (or anything but uniform) distribution, then guess what -- your sampling procedure's not so random after all.

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