 |
Download a pdf copy of this help file here |
|
Download a pdf copy of this help file here |
See also: VoseBinomialP
The confidence we have that the true value of the probability is less than or equal to p is:
P(X>s;n,p) + ½ P(X=s;n,p)
where X is the random variable of the number of successes one could have from n trials, and s is the number of observed successes. This is saying that the greater the true value of the probability p, the more confident we would be in observing a particular number of successes s, or more. Translated into Excel formulas, the confidence we have that the true probability is less than any specific tested value p is given by:
=1-VoseBinomialProb(s,n,p,1)+0.5*VoseBinomialProb(s,n,p,0)
We can use this to construct a cumulative confidence distribution:
Figure 1: Cumulative distributions of estimate of p for n = 10 trials and varying number of successes s
The points used to construct the above plot can be fed into a VoseCumulA distribution so that you can generate values from it.
However, ModelRisk offers the function VoseBinomialP that will perform this whole process automatically.
Look at the scenario for s = 0: the cumulative distribution starts with a value of 50% at p = 0 so it is saying that with no successes observed we have 50% confidence that there is no binomial process at all – trials can’t become successes, and the remaining 50% confidence is distributed over p = (0,1). The reverse logic applies where s = n.
In the VoseBinomialP function VoseBinomialP(s,n,ProcessExists,U) you input the successes s and trials n and in the situation where s = 0 or n you have the option to specify whether you know that the probability lies within (0,1) (ProcessExists = TRUE).
The U parameter also allows you to specify a cumulative percentile – if omitted the function simulates random values of what the value p might be. So, for example:
VoseBinomialP(10,20, TRUE, 0.99) = VoseBinomialP(10,20, FALSE, 0.99) = 0.74605
VoseBinomialP(0,20,TRUE,0.99) = 0.02522 (it assumes that p cannot be zero)
VoseBinomialP(0,20,FALSE,0.4) = 0 (it allows that p could be zero)