Where can i find historical data for the distribution ?
How does it fluctuate ? It is not constant, correct ?
Mr Nakamoto in his white paper assumed it was constant ?
The number of hashes a miner can do per second. What you seem to be referring to is Network Hashpower: the amount of computational power of all the miners on the network. This was not assumed to be constant, but the block time was intended to be close to constant, hence the reason the difficulty is adjusted based on how often blocks are mined.
How would you estimate the hashpower of the entire network without knowing all of the miners and details about their hardware, and how long they are running? You could potentially see the rate that shares are submitted to pools (and their difficulty) if you have access to that data. But, more likely you'll have to estimate it based on the current difficulty, and rate that blocks are solved. This can calculated based on the probability of finding a hash at the current target by guessing randomly, and then how many guesses per second (hashrate) it would take to get one every 10 minutes:
Hashrate = (1/P)/Blocktime, where
Pis the probability of guessing a hash that solves the block at the current target
Blocktimeis blocktime in seconds (600 seconds for bitcoin)
what distribution does the hashrate follow?
You can see that the difficulty follows the network hashrate. As the hashrate increases over time with more user adoption (miners coming), the difficulty increases to maintain a 10 minute block time.