I was literally just working on this problem! Although my main question was what is the expected value of x given some bound on the length and a set of probabilities.
When you say "p sub i" I just hear "peace of I" :) If you take {0,1,2} and fix p_1=0, the situation you describe as Cantor set, are you not still allowed to e.g. start with seven 1's and from then on only use 0's and 2's? So that the proportion goes to 0 as k goes to infinity, but you also represent w's which have a 1. Btw. with this limit, it the convergence of interest there? Do folks care how fast this gets close to p_j?
Yes! you are allowed to do such a thing (I obfuscate this in the video on purpose because it would take a bit too long to talk through the nuance there). Without getting into too many details, In the case of finitely many 1's you can think of this as having just another copy of the cantor set "somewhere else" in the interval. For having infinitely many 1's where the proportion is still 0 (think having an increasing unbounded sequence of gaps between 1's or something similar) there is a bit more to think about.
I was literally just working on this problem! Although my main question was what is the expected value of x given some bound on the length and a set of probabilities.
When you say "p sub i" I just hear "peace of I" :)
If you take {0,1,2} and fix p_1=0, the situation you describe as Cantor set, are you not still allowed to e.g. start with seven 1's and from then on only use 0's and 2's? So that the proportion goes to 0 as k goes to infinity, but you also represent w's which have a 1.
Btw. with this limit, it the convergence of interest there? Do folks care how fast this gets close to p_j?
Yes! you are allowed to do such a thing (I obfuscate this in the video on purpose because it would take a bit too long to talk through the nuance there).
Without getting into too many details, In the case of finitely many 1's you can think of this as having just another copy of the cantor set "somewhere else" in the interval. For having infinitely many 1's where the proportion is still 0 (think having an increasing unbounded sequence of gaps between 1's or something similar) there is a bit more to think about.
x_1