# Binomial distribution

The printable version is no longer supported and may have rendering errors. Please update your browser bookmarks and please use the default browser print function instead.

Bernoulli distribution

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

The probability distribution of a random variable $X$ which assumes integral values $x = 0 \dots n$ with the probabilities

$${\mathsf P} \{ X=x \} = b _ {x} (n, p) = \ \left ( \begin{array}{c} n \\ x \end{array} \right ) p ^ {x} (1-p) ^ {n-x} ,$$

where $( {} _ {x} ^ {n} )$ is the binomial coefficient, and $p$ is a parameter of the binomial distribution, called the probability of a positive outcome, which can take values in the interval $0 \leq p \leq 1$. The binomial distribution is one of the fundamental probability distributions connected with a sequence of independent trials. Let $Y _ {1} , Y _ {2} \dots$ be a sequence of independent random variables, each one of which may assume only one of the values 1 and 0 with respective probabilities $p$ and $1 - p$( i.e. all $Y _ {i}$ are binomially distributed with $n = 1$). The values of $Y _ {i}$ may be treated as the results of independent trials, with $Y _ {i} = 1$ if the result of the $i$- th trial is "positive" and $Y _ {i} = 0$ if it is "negative" . If the total number of independent trials $n$ is fixed, such a scheme is known as Bernoulli trials, and the total number of positive results,

$$X=Y _ {1} + \dots + Y _ {n} ,\ \ n \geq 1 ,$$

is then binomially distributed with parameter $p$.

The mathematical expectation ${\mathsf E} z ^ {X}$( the generating function of the binomial distribution) for any value of $z$ is the polynomial $[pz + (1 - p)] ^ {n}$, the representation of which by Newton's binomial series has the form

$$b _ {0} + b _ {1} z + \dots + b _ {n} z ^ {n} .$$

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

$${\mathsf E} X = np,$$

$${\mathsf D} X = {\mathsf E} (X-np) ^ {2} = np (1-p),$$

$${\mathsf E}(X-np) ^ {3} = np (1-p) (1 - 2p).$$

The binomial distribution function is defined, for any real $y$, $0 < y < n$, by the formula

$$F (y) = \ {\mathsf P} \{ X \leq u \} = \ \sum _ {x = 0 } ^ { [y] } \left ( \begin{array}{c} n \\ x \end{array} \right ) p ^ {x} (1 - p) ^ {n - x } ,$$

where $[y]$ is the integer part of $y$, and

$$F (y) \equiv \ \frac{1}{B([y] + 1, n - [y]) } \int\limits _ { p } ^ { 1 } t ^ {[y]} (1 - t) ^ {n - [y] - 1 } dt,$$

$B(a, b)$ is Euler's beta-function, and the integral on the right-hand side is known as the incomplete beta-function.

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

$$F (y) = \Phi \left [ \frac{y - np + 0.5 }{\sqrt {np (1 - p) } } \right ] + R _ {n} (y, p),$$

where

$$R _ {n} (y, p) = O (n ^ {-1/2 } )$$

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

If the number of independent trials $n$ is large, while the probability $p$ is small, the individual probabilities $b _ {x} (n, p)$ can be approximately expressed in terms of the Poisson distribution:

$$b _ {x} (n, p) = \ \left ( \begin{array}{c} n \\ x \end{array} \right ) p ^ {x} (1 - p) ^ {n - x } \approx \ \frac{(np) ^ {x} }{x!} e ^ {-np } .$$

If $n \rightarrow \infty$ and $0 < c \leq y \leq C$( where $c$ and $C$ are constants), the asymptotic formula

$$F (y) = \ \sum _ {x = 0 } ^ { [y] } \frac{\lambda ^ {x} }{x!} e ^ {- \lambda } + O (n ^ {-2} ),$$

where $\lambda = (2n - [y])p / (2 - p)$, is uniformly valid with respect to all $p$ in the interval $0 < p < 1$.

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

#### References

 [G] B.V. Gnedenko, "The theory of probability", Chelsea, reprint (1962) (Translated from Russian) [F] W. Feller, "An introduction to probability theory and its applications", Wiley (1957–1971) [PR] Yu.V. Prohorov, Yu.A. Rozanov, "Probability theory, basic concepts. Limit theorems, random processes" , Springer (1969) (Translated from Russian) MR0251754 [P] Yu.V. Prohorov, "Asymptotic behaviour of the binomial distribution" Selected Translations in Math. Stat. and Probab. , 1 , Amer. Math. Soc. (1961) MR0116370 (Translated from Russian) Uspekhi Mat. Nauk , 8 : 3 (1953) pp. 135–142 MR0056861 [BS] 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) MR0735434 Zbl 0529.62099
How to Cite This Entry:
Binomial distribution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Binomial_distribution&oldid=46067
This article was adapted from an original article by L.N. Bol'shev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article