View source for Binomial distribution

''Bernoulli distribution''

{{MSC|60E99}}

[[Category:Distribution theory]]

The probability distribution of a random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016420/b0164201.png" /> which assumes integral values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016420/b0164202.png" /> with the probabilities

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016420/b0164203.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016420/b0164204.png" /> is the binomial coefficient, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016420/b0164205.png" /> is a parameter of the binomial distribution, called the probability of a positive outcome, which can take values in the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016420/b0164206.png" />. The binomial distribution is one of the fundamental probability distributions connected with a sequence of independent trials. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016420/b0164207.png" /> be a sequence of independent random variables, each one of which may assume only one of the values 1 and 0 with respective probabilities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016420/b0164208.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016420/b0164209.png" /> (i.e. all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016420/b01642010.png" /> are binomially distributed with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016420/b01642011.png" />). The values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016420/b01642012.png" /> may be treated as the results of independent trials, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016420/b01642013.png" /> if the result of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016420/b01642014.png" />-th trial is "positive" and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016420/b01642015.png" /> if it is "negative" . If the total number of independent trials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016420/b01642016.png" /> is fixed, such a scheme is known as [[Bernoulli trials|Bernoulli trials]], and the total number of positive results,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016420/b01642017.png" /></td> </tr></table>

is then binomially distributed with parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016420/b01642018.png" />.

The mathematical expectation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016420/b01642019.png" /> (the [[Generating function|generating function]] of the binomial distribution) for any value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016420/b01642020.png" /> is the polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016420/b01642021.png" />, the representation of which by Newton's binomial series has the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016420/b01642022.png" /></td> </tr></table>

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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016420/b01642023.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016420/b01642024.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016420/b01642025.png" /></td> </tr></table>

The binomial distribution function is defined, for any real <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016420/b01642026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016420/b01642027.png" />, by the formula

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016420/b01642028.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016420/b01642029.png" /> is the integer part of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016420/b01642030.png" />, and

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016420/b01642031.png" /></td> </tr></table>

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016420/b01642032.png" /> is Euler's [[Beta-function|beta-function]], and the integral on the right-hand side is known as the incomplete beta-function.

As <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016420/b01642033.png" />, the binomial distribution function is expressed in terms of the standard normal distribution function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016420/b01642034.png" /> by the asymptotic formula (the de Moivre–Laplace theorem):

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016420/b01642035.png" /></td> </tr></table>

where

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016420/b01642036.png" /></td> </tr></table>

uniformly for all real <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016420/b01642037.png" />. There also exist other, higher order, normal approximations of the binomial distribution.

If the number of independent trials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016420/b01642038.png" /> is large, while the probability <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016420/b01642039.png" /> is small, the individual probabilities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016420/b01642040.png" /> can be approximately expressed in terms of the [[Poisson distribution|Poisson distribution]]:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016420/b01642041.png" /></td> </tr></table>

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016420/b01642042.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016420/b01642043.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016420/b01642044.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016420/b01642045.png" /> are constants), the asymptotic formula

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016420/b01642046.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016420/b01642047.png" />, is uniformly valid with respect to all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016420/b01642048.png" /> in the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016420/b01642049.png" />.

The [[Multinomial distribution|multinomial distribution]] is the multi-dimensional generalization of the binomial distribution.

====References====
{|
|valign="top"|{{Ref|G}}|| B.V. Gnedenko,   [[Gnedenko, "A course in the theory of probability"|"The theory of probability"]], Chelsea, reprint (1962)  (Translated from Russian)
|-
|valign="top"|{{Ref|F}}|| W. Feller,  [[Feller, "An introduction to probability theory and its    applications"|"An introduction to probability theory and its    applications"]], Wiley (1957–1971)
|-
|valign="top"|{{Ref|PR}}|| Yu.V.  Prohorov, Yu.A. Rozanov, "Probability theory, basic concepts. Limit theorems, random processes" , Springer (1969) (Translated from Russian) {{MR|0251754}} {{ZBL|}}
|-
|valign="top"|{{Ref|P}}|| Yu.V. Prokhorov, "Asymptotic behaviour of the binomial distribution" ''Selected Translations in Math. Stat. and Probab.'' , '''1''' , Amer. Math. Soc. (1961) {{MR|0116370}} (Translated from Russian) ''Uspekhi Mat. Nauk'' , '''8''' : 3 (1953) pp. 135–142 {{MR|0056861}}
|-
|valign="top"|{{Ref|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) {{MR|0735434}} {{ZBL|0529.62099}}
|}