Motivation
The binomial distribution is quite common as it models chain of independent events each with binary distribution, it can also be used when each binary event has different probability this gives the ability to support iterative nature of events. However, the problem arises when we look on the whole picture of n events, in this case the variance is fixed without ability to control it, unlike normal distribution.
The question is how can we remain the iterative nature of the binomial distribution while controlling the variance of the total n events?
The general idea is to change the probability of the event based on the history. For example, if p=0.6 in the first even we use p0=0.6, and let's say we got 0 so in the next event we'll use a raise the probability, say p1=0.66, and if we got another 0 we'll continue to raise the probability, for example, p2=0.69. And the other way around, if we get 1 we will reduce the probability to get 1, so one can summarize that we take chances from the unlikely high to the unlikely low, and this is where the probability got its name from (it is important to add that increasing the variance is also possible when going the other way around)
The challenges
We need to perform these changes in the probabilities in a manner that we would define a distribution with expected values and variance, SB(n,p,v). The expected value should equal to binomial distribution, E(B(n,p)) = E(SB(n,p,v)) = np. The variance will be extension of the binomial variance (V(B(n,p)) = npq) and will be defined as V(SB(n,p,v)) = pq*f(n,v).
Solution Overview
Instead of using a constant p for all events we will define p(n,k)=p+(k-np)*v.
Using this probabilities for each event we will get that expectation is the same and the variance will be defined are Var(SB(n,p,v)) = sum((1-2v)^n) = ((1-2v)^n+1)/2v.
It is interesting to see that (1+2v)^n is a binomial series sum.
Results
Here are the graphs of the SB(100,6,v) where v goes from -0.02 : 0.015
It seems like very nice Gaussians as expected from binomial distributions and we can see that the variance is controlled.
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