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''Bernoulli distribution''
 
''Bernoulli distribution''
  
Line 5: Line 17:
 
[[Category:Distribution theory]]
 
[[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
+
The probability distribution of a random variable $  X $
 +
which assumes integral values $  x = 0 \dots n $
 +
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>
+
$$
 +
{\mathsf P} \{ X=x \}  = b _ {x} (n, p)  = \
 +
\left ( \begin{array}{c}
 +
n \\
 +
x
 +
\end{array}
 +
\right ) p  ^ {x} (1-p)  ^ {n-x} ,
 +
$$
  
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,
+
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|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>
+
$$
 +
X=Y _ {1} + \dots +
 +
Y _ {n} ,\ \
 +
n \geq  1 ,
 +
$$
  
is then binomially distributed with parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016420/b01642018.png" />.
+
is then binomially distributed with parameter $  p $.
  
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
+
The mathematical expectation $  {\mathsf E} z  ^ {X} $(
 +
the [[Generating function|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
  
<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>
+
$$
 +
b _ {0} + b _ {1} z + \dots + b _ {n} z  ^ {n} .
 +
$$
  
 
(Hence the very name "binomial distribution" .) The moments (cf. [[Moment|Moment]]) of a binomial distribution are given by the formulas
 
(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>
+
$$
 +
{\mathsf E} X  = np,
 +
$$
 +
 
 +
$$
 +
{\mathsf D} X  = {\mathsf E} (X-np)  ^ {2}  = np (1-p),
 +
$$
 +
 
 +
$$
 +
{\mathsf E}(X-np)  ^ {3}  = np (1-p) (1 - 2p).
 +
$$
 +
 
 +
The binomial distribution function is defined, for any real  $  y $,
 +
$  0 < y < n $,
 +
by the formula
 +
 
 +
$$
 +
F (y)  = \
 +
{\mathsf P} \{ X \leq  u \}  = \
 +
\sum _ {x = 0 } ^ { [y] }
 +
\left ( \begin{array}{c}
 +
n \\
 +
x
 +
\end{array}
 +
\right ) p  ^ {x} (1 - p) ^ {n - x } ,
 +
$$
  
<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>
+
where  $  [y] $
 +
is the integer part of  $  y $,
 +
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/b01642025.png" /></td> </tr></table>
+
$$
 +
F (y)  \equiv \
  
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
+
\frac{1}{B([y] + 1, n - [y]) }
  
<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>
+
\int\limits _ { p } ^ { 1 }
 +
t  ^ {[y]}
 +
(1 - t) ^ {n - [y] - 1 }  dt,
 +
$$
  
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
+
$  B(a, b) $
 +
is Euler's [[Beta-function|beta-function]], and the integral on the right-hand side is known as the incomplete beta-function.
  
<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>
+
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):
  
<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.
+
$$
 +
F (y)  = \Phi
 +
\left [
  
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):
+
\frac{y - np + 0.5 }{\sqrt {np (1 - p) } }
  
<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>
+
\right ] +
 +
R _ {n} (y, p),
 +
$$
  
 
where
 
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>
+
$$
 +
R _ {n} (y, p)  = O (n ^ {-1/2 } )
 +
$$
 +
 
 +
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|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!}
  
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.
+
e ^ {-np } .
 +
$$
  
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]]:
+
If $  n \rightarrow \infty $
 +
and  $  0 < c \leq  y \leq  C $(
 +
where  $  c $
 +
and  $  C $
 +
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/b01642041.png" /></td> </tr></table>
+
$$
 +
F (y)  = \
 +
\sum _ {x = 0 } ^ { [y] }
  
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
+
\frac{\lambda  ^ {x} }{x!}
  
<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>
+
e ^ {- \lambda } + O (n  ^ {-2} ),
 +
$$
  
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" />.
+
where $  \lambda = (2n - [y])p / (2 - p) $,  
 +
is uniformly valid with respect to all $  p $
 +
in the interval $  0 < p < 1 $.
  
 
The [[Multinomial distribution|multinomial distribution]] is the multi-dimensional generalization of the binomial distribution.
 
The [[Multinomial distribution|multinomial distribution]] is the multi-dimensional generalization of the binomial distribution.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  B.V. Gnedenko,  [[Gnedenko, "A course in the theory of probability"|"The theory of probability"]], Chelsea, reprint (1962)  (Translated from Russian) </TD></TR>
+
{|
<TR><TD  valign="top">[2]</TD> <TD valign="top"> W. Feller,  [[Feller, "An introduction to probability theory and its    applications"|"An introduction to probability theory and its    applications"]], Wiley (1957–1971)</TD></TR>
+
|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)
<TR><TD valign="top">[3]</TD> <TD valign="top"> Yu.V.  Prohorov [Prokhorov], Yu.A. Rozanov, "Probability theory, basic concepts. Limit theorems, random processes" , Springer (1969) (Translated from Russian) {{MR|0251754}} {{ZBL|}} </TD></TR>
+
|-
<TR><TD valign="top">[4]</TD> <TD valign="top"> 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}} </TD></TR>
+
|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)
<TR><TD valign="top">[5]</TD> <TD valign="top"> 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}} </TD></TR></table>
+
|-
 +
|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. Prohorov, "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}}
 +
|}

Latest revision as of 10:59, 29 May 2020


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=25816
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