A Random Variable ‘x’ follows the binomial distribution having the parameters ‘n’ and ‘p’ then the probability mass function can be written as,
g(x; n, p) = P(X = x) = n! / x! (n – x)! px (1 - p)n-x,
This is the binomial coefficient.
Cumulative Binomial Distribution can be written mathematically as,
g(x; n, p) = P(X ≤ x) = i = 0∑⌊x⌋n! / k! (n – k)! pk (1 – p)n – k,
Where symbol '⌊x⌋' refers the floor under ‘x’ that Mean the greatest Integer is either less or equals to zero.
This expression also can be represented in the terms of the regularized incomplete beta function and written as
g(x; n, p) = P(X ≤ x) = I 1 – p (n – x, x + 1),
g(x; n, p) = (n – x) n! / x! (n – x)! 0∫1 – p t (n - x – 1) (1 – t)x dt,
For the value x ≤ np, upper bounds for the lower tail of the distribution function can be derived easily as Hoeffding's inequality yields the bound.
G(x; n, p) ≤ ( ½ ) exp [-2 (np – x)2 / n]
And the Chernoff's inequality can used to determine this bound
G(x; n, p) ≤ exp [(-1/2p) (np – x)2 / n],
The bounds would be reasonably tight if probability ‘p’ has the value (½),
G(x; n, (½)) ≥ (1/15) exp [-16 (n/2) – x)2 / n],
Since the above expression is valid for all x ≥ 3n / 8.