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If you repeatedly add 2 and then divide by 2, you approach the number 2. If you repeatedly flip a coin to either add 2 or divide by 2, what distribution do you
by jon_richards 6y ago
If you repeatedly add 2 and then divide by 2, you approach the number 2.
If you repeatedly flip a coin to either add 2 or divide by 2, what distribution do you approach?
If you have a distribution, what random processes are identities for that distribution? Which are stable? Which will reproduce the distribution from any starting distribution?
It’s related to matrix kernels, but it’s hard to generalize to continuous numbers. I spent a while looking into it years ago and couldn’t find much.
My goal was to eventually take it a step farther and create simple stochastic functions for certain behaviors, such as bistable switches, memory registers, etc.
- mcphage 6y agoI would guess that you still get to 2—because the more times you add 2, the greater the impact of dividing by 2, and the more times you divide by 2, the greater the impact of adding 2. Is that the case?
- jamesmaniscalco 6y agoIf you run the random process many times and then look at a histogram of the vector of outputs, the distribution peaks somewhat higher than 2 [0], but the output of the random process does not tend towards any particular number or sequence of numbers. Indeed, the distribution looks highly discontinuous and fractal. It's a very intriguing question. I suspect thinking in terms of rings (abstract algebra) would be helpful. [0] "somewhat higher" - I think this is because the sequence "add two then divide by two" gives you back your input when you start with 2, while "divide by two then add two" does when you start with 4; the distribution should peak between these two "stable" points. Exactly where the distribution peaks probably depends on your histogram bin width.
- mcphage 6y agoHuh—that is interesting!