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Here 'truncating' implies reducing precision of the random numbers and not truncating the series of random numbers. For example, if I have $n$ truly random numbers (drawn from any distribution, e.g., normal, uniform, etc.) with arbitrary precision and I truncate all the numbers so that finally I end up with a set of $n$ numbers, each with exactly 2 digits after the decimal. Can I call this new set of numbers 'random'?

I came up with this question when I was reading about hardware generated random numbers. The Wikipedia article says they generate random numbers by measuring a physical process. But since this measurement has its limitations (measurement error, finite precision, etc) can we call these hardware generated numbers random?

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see also here –  user603 Jul 22 '12 at 22:13

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up vote 18 down vote accepted

Yes, the truncated values are random. The distribution has changed from a continuous distribution to a discrete distribution. Random values with discrete distributions are often used.

There are senses in which this change to the distribution is very small. The maximum difference between the cumulative distribution functions is bounded by the maximum density of the original times the maximum change from rounding. The changes to the expected value and the standard deviation can be bounded similarly.

There are senses in which the change to the distribution is large, e.g., the $L_1$ distance between any continuous distribution and any discrete distribution is maximal. The average of some discontinuous functions of the value may change a lot.

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This answer is wrong. Unfortunately @steadyfish has asked for "any distribution". Let $X$ be the result of rolling a die. Then append 1.00 to the front of $X$, so that the sample space is 1.001, 1.002, 1.003, and so on. Truncating to two digits would leave you with a degenerate random variable. –  nomen Aug 28 at 17:44

Douglas' answer is completely correct, fixed precision random numbers are still random but with a different distribution than the one intended at the beginning. Since all uniform random generators in a software like R or on a hardware generator like Intel's produce fixed precision outcomes, from a formal point of view, they are all "wrong" in that they do not return a random value on (0,1), but a pseudo-random value on a finite set. Again, Douglas' point is quite correct that this approximation error is mostly harmless.

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+1, in line with this, one additional point is that numbers generated by computer algorithms, such as R & other statistical software use, are actually pseudorandom. They will ultimately circle back to the same values / can be made repeatable by setting the seed, whereas I believe this is not true of hardware generators. –  gung Jul 23 '12 at 15:07
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Indeed, thanks, I corrected random into pseudorandom. Having a (huge) period is actually not the only reason why pseudorandom generators are not producing random numbers, since they are deterministic. Using twice the same seed leads twice to the same sequence. –  Xi'an Jul 24 '12 at 21:24
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...which is actually a good thing, since it allows us to reproduce the results of simulation studies. –  MånsT Jul 25 '12 at 6:07
    
I must disagree. You can easily end up with a degenerate distribution by truncating. It is difficult to call the result "random". –  nomen Aug 28 at 17:46

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