Corrections to the two-sided probability and hypothesis test statistics on binomial distributions

Binomial distributions widely exist in nature and human society,which is classified as discrete by probability theory.In theoretical studies in mathematical statistics,a random variable of binomial distribution B(n,p) is equivalent to the sum of a number of n independent and identical variables of Bernoulli distribution B(1,p).Estimation and testing on parameter p of binomial distribution B(n,p) is therefore equivalent to those of Bernoulli distribution B(1,p).Three corrections were made in this article,relevant to the calculation of two-tailed probability,and the construction of hypothesis test statistics.(1) Assume pk(k=0,1,…,n) is the probability list of binomial distribution B(n,p),and the probability by ascending order is given by p(k).The two-tailed exact probability is equal to ■,given the value of the observed k.(2) When testing the difference between parameter p of B(n,p) against a given value p0,the test statistic was corrected by ■,which asymptotically approaches to normal distribution N(p-p0,1) un