## Binomial Distribution

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 Moment-Generating Function for the distribution is
 (5) (6) (7)

The Mean is
 (8)

 (9) (10) (11) (12)

so the Moments about the Mean are
 (13) (14) (15)

The Skewness and Kurtosis are
 (16) (17)

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
 (23) (24)

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
 (30) (31) (32)

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
 (36)

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!

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, F-Distribution, Cumulative Binomial Distribution.'' §6.2 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 219-223, 1992.

Spiegel, M. R. Theory and Problems of Probability and Statistics. New York: McGraw-Hill, p. 108-109, 1992.