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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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    $\begingroup$ see also here $\endgroup$ – user603 Jul 22 '12 at 22:13
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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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    $\begingroup$ 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. $\endgroup$ – nomen Aug 28 '14 at 17:44

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