Will a Random Number Generator Eventually Repeat Its Numbers?

[CRGreathouse]

Ok here is what I have for a normal distribution "being normal means that any finite string of digits occurs as often as expected, that is, 1234 makes up 1/10000 of the four-digit strings, etc". Is that what you mean 'normal' in a statistical sense? What would a "uniform distribution" be? If it is one with a equal number of each digit? In that case a "uniform" distribution would just be a particular requirement of a "normal" distribution. I don't see the logic of why a uniform random infinite sequence that would be a "uniform" distribution would not also be a "normal" distribution.

The rational 123456789/9999999999 has exactly (in an asymptotic sense) 1/10 of its digits as 0, 1, ..., 9, but it's not normal because 11 doesn't appear in its decimal expansion.

 

[ramsey2879]

The rational 123456789/9999999999 has exactly (in an asymptotic sense) 1/10 of its digits as 0, 1, ..., 9, but it's not normal because 11 doesn't appear in its decimal expansion.

I agree that all uniform distributions are not necessarily normal but I am saying that if one holds that a infinite random sequence would be a uniform sequence then by the same logic that the number 9 is 1/10 of the random sequence then 11 would be 1/100 of the random sequence since the logic would not depend upon what number base you are working with in that case.

 

[SW VandeCarr]

I agree that all uniform distributions are not necessarily normal but I am saying that if one holds that a infinite random sequence would be a uniform sequence then by the same logic that the number 9 is 1/10 of the random sequence then 11 would be 1/100 of the random sequence since the logic would not depend upon what number base you are working with in that case.

There is apparently more than one usage of 'normal distribution' here. In statistics the term refers to the Gaussian probability distribution. In a uniform probability distribution each outcome has the same probability. This is not the case in a Gaussian distribution.

 

[ramsey2879]

There is apparently more than one usage of 'normal distribution' here. In statistics the term refers to the Gaussian probability distribution. In a uniform probability distribution each outcome has the same probability. This is not the case in a Gaussian distribution.

Doesn't "11" have the "same probability" that any other two digit number? Doesn't the number "123456789" have the same probability as any other 9 digit number? Then what is the difference between saying that 9 occurs 1/10 of the time in an infinite random sequence because each digit has an equal probability from saying that "1234" occurs 1/10000 th of the time because each 4 digit number has the same probability?

 

[CRGreathouse]

There is apparently more than one usage of 'normal distribution' here. In statistics the term refers to the Gaussian probability distribution. In a uniform probability distribution each outcome has the same probability. This is not the case in a Gaussian distribution.

Normal *number*, not normal *distribution*. No one here is talking about the distribution produced by the Central Limit Theorem.

 

[ramsey2879]

Normal *number*, not normal *distribution*. No one here is talking about the distribution produced by the Central Limit Theorem.

Thanks, I studied the information re central central limit therom and agree that my terminology was wrong. The central limit therom applies when there are a large number of finite samples of "normal" numbers (each having the same probability of occurring within the applicable range) is taken and the mean of each sample is taken. Then the distribution of the means will approach a normal distribution.
But getting back to my understanding of a uniform distribution, if each digit has an equal probability has a 1/10th probility, then each 2 digit number has a 1/100th probability etc., and by the same argument that a infinite random sequence of one digit numbers will have a uniform distribution then the same sequence cam be considered as an infinite sequence of n-digit numbers, the distribution of which will be a uniform distribution.

 

[CRGreathouse]

The relevant answer would be that a number generated from the suitable uniform distribution of digits is normal with probability 1.