Sampling from a continuous distribution

Question

The Lebesgue integral of the standard normal pdf over $\mathbb{Q}$ is equal to zero, since the rationals are countable and thus have measure zero. So the probability of "drawing" a rational number from the standard normal is equal to zero. You are guaranteed to draw an irrational.

In applied statistics, however, "sampling" from the standard normal using a random number generator only yields rational numbers that can be represented exactly under the IEEE-754 standard. This also limits the range of possible values.

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Expectation

1. Rationals should be impossible to draw.
2. We draw from a continuous distribution.
3. Range is all of $\mathbb{R}$.

Reality

1. Rationals are all we get.
2. We draw from a discrete distribution.
3. Range is a subset of $\left[10^{-309}, 10^{309}\right] $ (with double precision).

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Question

What does this imply for applied stats, in particular when random samples from continuous distributions are simulated?

Answer 1

In applied stats it's not a problem since the rational numbers are dense in $\mathbb{R}$. The difference between the set of rational numbers and any real number is $0$, since you can always find a rational number $q$ arbitrarily close to any irrational number $r$. So although we're sampling from the rationals, since the rationals are dense in $\mathbb{R}$, this has no real consequences to us.

I would liken this issue to that of "true randomness." In reality, true randomness cannot be generated in any real sense, and so "truly random samples" cannot be generated. But pseudo-random number generators are plentiful and basically indistinguishable from true random number generators in any practical setting.

Answer 2

One implication is that the tails after a cutoff are not represented at all when "sampling from the normal", so "extremely extreme" events are absent from simulations.