# How random is the digits of $\pi$? [duplicate]

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Distribution of the digits of Pi

If we plot the digits of $\pi$

$3.1415926535897932384626433832795028841971693993751058\cdots$

versus the indices

$((1, 3), (2, 1), (3, 4), (4, 1), \cdots)$

How to quantify the randomness of the resultant curve?

The first part of the curve looks like this .

### Re: @Henning Makholm

Rephrasing the question, suppose the curve happens to be a sample function of a random process, how to give the distribution of that random process?

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In one quite important sense they are not random at all, because they are the one and only sequence that are the decimal digits of $\pi$. That's an extremely specific property. –  Henning Makholm Dec 15 '12 at 16:30
You might like to take a look at Normal Numbers. –  Old John Dec 15 '12 at 16:35
There are lots of ways to try to quantify randomness. Which are appropriate will depend on what you intend to do with the result. –  Chris Eagle Dec 15 '12 at 16:35
Regarding your edit: please be a bit more specific when throwing around terms such as 'random process' and 'distribution'. For example a distribution is either discrete or continuous - which is it here? It also has a mean - what is the mean in this case? Also please do have a look at @OldJohn's link, before stabbing in the dark more... –  Peter Sheldrick Dec 15 '12 at 16:52
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## marked as duplicate by Asaf Karagila, Did, draks ..., Quixotic, Davide GiraudoDec 15 '12 at 21:31

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

## 1 Answer

Histogram of the first 10000 digits

Covariance of the first 10000 consecutive digits: -0.09 (negligible)

Mathematica code used:

Histogram:

Histogram[RealDigits[Pi, 10, 10000][[1]], 10]


Covariance: (I'm sure there's a more efficient way to do this)

N[Covariance[RealDigits[Pi, 10, 10000][[1]][[Range[1, 9999]]],
RealDigits[Pi, 10, 10000][[1]][[Range[2, 10000]]]]]

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is that a wolfram alpha plot? –  Peter Sheldrick Dec 15 '12 at 17:02
@PeterSheldrick: mathematica: Histogram[RealDigits[Pi, 10, 10000][[1]], 10] –  akkkk Dec 15 '12 at 17:02
If any of you guys have more requests, I'll be happy to compute/plot more data. –  akkkk Dec 15 '12 at 17:04
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