The formula can be understood as follows. Note that uniform normal exponential poisson binomial pdf probability of it occurring can be fairly small.
What is the probability of achieving 0, 1,, 6 heads after six tosses? 4, then the average number of successful results will be 25. 2 is the unique median. This result was first derived by Katz et al in 1978. No closed form for this distribution is known. An asymptotic approximation for the mean is known. This 3-standard-deviation rule is equivalent to the following conditions, which also imply the first rule above.
However, the specific number varies from source to source, and depends on how good an approximation one wants. In particular, if one uses 9 instead of 5, the rule implies the results stated in the previous paragraphs. The proportion of people who agree will of course depend on the sample. Because of this problem several methods to estimate confidence intervals have been proposed. Secondly, this formula does not use a plus-minus to define the two bounds.
The Wald method, although commonly recommended in textbooks, is the most biased. One way to generate random samples from a binomial distribution is to use an inversion algorithm. These probabilities should sum to a value close to one, in order to encompass the entire sample space. 0 and 1, one can transform the calculated samples U into discrete numbers by using the probabilities calculated in step one. Median der Binomial- and Poissonverteilung”. Mean, Median and Mode in Binomial Distributions”.