Binomial Distribution
المؤلف:
Beyer, W. H.
المصدر:
CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press
الجزء والصفحة:
...
17-4-2021
3964
Binomial Distribution
The binomial distribution gives the discrete probability distribution 
of obtaining exactly 
successes out of 
Bernoulli trials (where the result of each Bernoulli trial is true with probability 
and false with probability 
). The binomial distribution is therefore given by
where 
is a binomial coefficient. The above plot shows the distribution of 
successes out of 
trials with 
.
The binomial distribution is implemented in the Wolfram Language as BinomialDistribution[n, p].
The probability of obtaining more successes than the 
observed in a binomial distribution is
 |
(3)
|
where
 |
(4)
|
is the beta function, and 
is the incomplete beta function.
The characteristic function for the binomial distribution is
 |
(5)
|
(Papoulis 1984, p. 154). The moment-generating function 
for the distribution is
The mean is
The moments about 0 are
so the moments about the mean are
The skewness and kurtosis excess are
The first cumulant is
 |
(26)
|
and subsequent cumulants are given by the recurrence relation
 |
(27)
|
The mean deviation is given by
 |
(28)
|
For the special case 
, this is equal to
where 
is a double factorial. For 
, 2, ..., the first few values are therefore 1/2, 1/2, 3/4, 3/4, 15/16, 15/16, ... (OEIS A086116 and A086117). The general case is given by
 |
(31)
|
Steinhaus (1999, pp. 25-28) considers the expected number of squares 
containing a given number of grains 
on board of size 
after random distribution of 
of grains,
 |
(32)
|
Taking 
gives the results summarized in the following table.
 |
 |
| 0 |
23.3591 |
| 1 |
23.7299 |
| 2 |
11.8650 |
| 3 |
3.89221 |
| 4 |
0.942162 |
| 5 |
0.179459 |
| 6 |
0.0280109 |
| 7 |
0.0036840 |
| 8 |
 |
| 9 |
 |
| 10 |
 |
An approximation to the binomial distribution for large 
can be obtained by expanding about the value 
where 
is a maximum, i.e., where 
. Since the logarithm function is monotonic, we can instead choose to expand the logarithm. Let 
, then
![ln[P(n)]=ln[P(n^~)]+B_1eta+1/2B_2eta^2+1/(3!)B_3eta^3+...,](https://mathworld.wolfram.com/images/equations/BinomialDistribution/NumberedEquation9.gif) |
(33)
|
where
![B_k=[(d^kln[P(n)])/(dn^k)]_(n=n^~).](https://mathworld.wolfram.com/images/equations/BinomialDistribution/NumberedEquation10.gif) |
(34)
|
But we are expanding about the maximum, so, by definition,
![B_1=[(dln[P(n)])/(dn)]_(n=n^~)=0.](https://mathworld.wolfram.com/images/equations/BinomialDistribution/NumberedEquation11.gif) |
(35)
|
This also means that 
is negative, so we can write 
. Now, taking the logarithm of (◇) gives
![ln[P(n)]=lnN!-lnn!-ln(N-n)!+nlnp+(N-n)lnq.](https://mathworld.wolfram.com/images/equations/BinomialDistribution/NumberedEquation12.gif) |
(36)
|
For large 
and 
we can use Stirling's approximation
 |
(37)
|
so
and
![(dln[P(n)])/(dn) approx -lnn+ln(N-n)+lnp-lnq.](https://mathworld.wolfram.com/images/equations/BinomialDistribution/NumberedEquation14.gif) |
(43)
|
To find 
, set this expression to 0 and solve for 
,
 |
(44)
|
 |
(45)
|
 |
(46)
|
 |
(47)
|
since 
. We can now find the terms in the expansion
Now, treating the distribution as continuous,
 |
(60)
|
Since each term is of order 
smaller than the previous, we can ignore terms higher than 
, so
 |
(61)
|
The probability must be normalized, so
 |
(62)
|
and
Defining 
,
![P(n)=1/(sigmasqrt(2pi))exp[-((n-n^~)^2)/(2sigma^2)],](https://mathworld.wolfram.com/images/equations/BinomialDistribution/NumberedEquation22.gif) |
(65)
|
which is a normal distribution. The binomial distribution is therefore approximated by a normal distribution for any fixed 
(even if 
is small) as 
is taken to infinity.
If 
and 
in such a way that 
, then the binomial distribution converges to the Poisson distribution with mean 
.
Let 
and 
be independent binomial random variables characterized by parameters 
and 
. The conditional probability of 
given that 
is
 |
(66)
|
Note that this is a hypergeometric distribution.
REFERENCES:
Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 531, 1987.
Papoulis, A. Probability, Random Variables, and Stochastic Processes, 2nd ed. New York: McGraw-Hill, pp. 102-103, 1984.
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. "Incomplete Beta Function, Student's Distribution, F-Distribution, Cumulative Binomial Distribution." §6.2 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 219-223, 1992.
Spiegel, M. R. Theory and Problems of Probability and Statistics. New York: McGraw-Hill, pp. 108-109, 1992.
Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, 1999.
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