The probability of successes in Bernoulli Trials is

(1) 
The probability of obtaining more successes than the observed is

(2) 
where

(3) 
is the Beta Function, and is the incomplete Beta Function. The Characteristic
Function is

(4) 
The MomentGenerating Function for the distribution is
The Mean is

(8) 
The Moments about 0 are
so the Moments about the Mean are
The Skewness and Kurtosis are
An approximation to the Bernoulli distribution for large can be obtained by expanding about the value where
is a maximum, i.e., where . Since the Logarithm function is Monotonic, we can instead choose to expand the Logarithm. Let
, then

(18) 
where

(19) 
But we are expanding about the maximum, so, by definition,

(20) 
This also means that is negative, so we can write . Now, taking the Logarithm of (1) gives

(21) 
For large and we can use Stirling's Approximation

(22) 
so
and

(25) 
To find , set this expression to 0 and solve for ,

(26) 

(27) 

(28) 

(29) 
since . We can now find the terms in the expansion
Now, treating the distribution as continuous,

(33) 
Since each term is of order
smaller than the previous, we can ignore terms higher than , so

(34) 
The probability must be normalized, so

(35) 
and
Defining
,

(37) 
which is a Gaussian Distribution. For , a different approximation procedure shows that the binomial
distribution approaches the Poisson Distribution. The first Cumulant is

(38) 
and subsequent Cumulants are given by the Recurrence Relation

(39) 
Let and be independent binomial Random Variables characterized by parameters and . The
Conditional Probability of given that is



(40) 
Note that this is a Hypergeometric Distribution!
See also de MoivreLaplace Theorem, Hypergeometric Distribution, Negative Binomial Distribution
References
Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 531, 1987.
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T.
``Incomplete Beta Function, Student's Distribution, FDistribution, Cumulative Binomial Distribution.'' §6.2 in
Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England:
Cambridge University Press, pp. 219223, 1992.
Spiegel, M. R. Theory and Problems of Probability and Statistics. New York: McGrawHill, p. 108109, 1992.
© 19969 Eric W. Weisstein
19990526