# normal distribution intuition; bits in a bit string ~ binomial distribution

I've been thinking about the intuition behind the normal distribution.

I think I know how the Central Limit Theorem effectively explains why so many things in nature end up being normally distributed: it's common that lots of variable (such as human height) are the result of a combination or summation of sorts of many other random variables following various distributions, and the CLM says that that summation is in turn normally distributed as the number of variables combined (summed) increases.

I've also started on a new line of thinking involving the binomial distribution (which is of course closely related to the normal distribution). Hear me out. The information needed to describe some natural property can be encoded as a bit string of 1s and 0s. Assuming the bit string has length n, and each bit is 0 or 1 with equal probability, the bit string follows the binomial distribution in a sense and as n increases in effect the normal distribution. So, it seems then that the underlying "essence" information itself to describe a given property can be seen to be linked to the normal distribution in this regard.

I hope you can follow my thinking. Does this make any sense? Probably there is a formal, mathematical theory which is similar to what I'm thinking?

• "the bit string follows the binomial distribution" No, what follows a binomial distribution is the "weight" (number of ones) of the bit string - and that, assuming the length is fixed and that all bit strings are equiprobable. Apr 14 at 13:23
• You're right. I'm being liberal with the language but I hope you see my point.
– Alex
Apr 14 at 20:25
• I am not entirely sure what your question is... If you are after some intuition as to why the Normal density looks the way it does, you might like this answer, which I tried to make "intuitive". If, on the other hand, you simply wish to validate your own intuition regarding bits, information, and the Normal density, I would suggest looking at the concept of entropy. In your case, I am not entirely sure your intuition is valid since, as noted by @leonbloy, what the Normal "records" in your example is an aggregation of that information. Apr 15 at 16:25