Binomial distribution

Bernoulli distribution

2020 Mathematics Subject Classification: Primary: 60E99 [MSN][ZBL]

The probability distribution of a random variable which assumes integral values with the probabilities

where is the binomial coefficient, and is a parameter of the binomial distribution, called the probability of a positive outcome, which can take values in the interval . The binomial distribution is one of the fundamental probability distributions connected with a sequence of independent trials. Let be a sequence of independent random variables, each one of which may assume only one of the values 1 and 0 with respective probabilities and (i.e. all are binomially distributed with ). The values of may be treated as the results of independent trials, with if the result of the -th trial is "positive" and if it is "negative" . If the total number of independent trials is fixed, such a scheme is known as Bernoulli trials, and the total number of positive results,

is then binomially distributed with parameter .

The mathematical expectation (the generating function of the binomial distribution) for any value of is the polynomial , the representation of which by Newton's binomial series has the form

(Hence the very name "binomial distribution" .) The moments (cf. Moment) of a binomial distribution are given by the formulas

The binomial distribution function is defined, for any real , , by the formula

where is the integer part of , and

is Euler's beta-function, and the integral on the right-hand side is known as the incomplete beta-function.

As , the binomial distribution function is expressed in terms of the standard normal distribution function by the asymptotic formula (the de Moivre–Laplace theorem):

where

uniformly for all real . There also exist other, higher order, normal approximations of the binomial distribution.

If the number of independent trials is large, while the probability is small, the individual probabilities can be approximately expressed in terms of the Poisson distribution:

If and (where and are constants), the asymptotic formula

where , is uniformly valid with respect to all in the interval .

The multinomial distribution is the multi-dimensional generalization of the binomial distribution.

References

[1]	B.V. Gnedenko, "The theory of probability" , Chelsea, reprint (1962) (Translated from Russian) MR0149513 Zbl 0102.34402
[2]	W. Feller, "An introduction to probability theory and its applications" , 1–2 , Wiley (1957–1971) MR0779091 MR0779090 MR0270403 MR0228020 MR1534302 MR0243559 MR0242202 MR0210154 MR1570945 MR0088081 MR1528130 MR0067380 MR0038583 Zbl 0598.60003 Zbl 0598.60002 Zbl 0219.60003 Zbl 0155.23101 Zbl 0158.34902 Zbl 0151.22403 Zbl 0138.10207 Zbl 0115.35308 Zbl 0077.12201 Zbl 0039.13201
[3]	Yu.V. [Yu.V. Prokhorov] Prohorov, Yu.A. Rozanov, "Probability theory, basic concepts. Limit theorems, random processes" , Springer (1969) (Translated from Russian) MR0251754
[4]	Yu.V. Prokhorov, "Asymptotic behaviour of the binomial distribution" Selected Translations in Math. Stat. and Probab. , 1 , Amer. Math. Soc. (1961) (Translated from Russian) Uspekhi Mat. Nauk , 8 : 3 (1953) pp. 135–142
[5]	L.N. Bol'shev, N.V. Smirnov, "Tables of mathematical statistics" , Libr. math. tables , 46 , Nauka (1983) (In Russian) (Processed by L.S. Bark and E.S. Kedrova) Zbl 0529.62099