Binomial distribution

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