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Binomial Distribution

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The binomial distribution gives the discrete probability distribution P_p(n|N) of obtaining exactly n successes out of N Bernoulli trials (where the result of each Bernoulli trial is true with probability p and false with probability q=1-p). The binomial distribution is therefore given by

P_p(n|N)=(N; n)p^nq^(N-n)
(1)
=(N!)/(n!(N-n)!)p^n(1-p)^(N-n),
(2)

where (N; n) is a binomial coefficient. The above plot shows the distribution of n successes out of N=20 trials with p=q=1/2.

The binomial distribution is implemented in the Wolfram Language as BinomialDistribution[n, p].

The probability of obtaining more successes than the n observed in a binomial distribution is

P=sum_(k=n+1)^N(N; k)p^k(1-p)^(N-k)=I_p(n+1,N-n),
(3)

where

I_x(a,b)=(B(x;a,b))/(B(a,b)),
(4)

B(a,b) is the beta function, and B(x;a,b) is the incomplete beta function.

The characteristic function for the binomial distribution is

phi(t)=(q+pe^(it))^N
(5)

(Papoulis 1984, p. 154). The moment-generating function M for the distribution is

M(t)=<e^(tn)>
(6)
=sum_(n=0)^(N)e^(tn)(N; n)p^nq^(N-n)
(7)
=sum_(n=0)^(N)(N; n)(pe^t)^n(1-p)^(N-n)
(8)
=[pe^t+(1-p)]^N
(9)
M^'(t)=N[pe^t+(1-p)]^(N-1)(pe^t)
(10)
M^('')(t)=N(N-1)[pe^t+(1-p)]^(N-2)(pe^t)^2+N[pe^t+(1-p)]^(N-1)(pe^t).
(11)

The mean is

mu=M^'(0)
(12)
=N(p+1-p)p
(13)
=Np.
(14)

The moments about 0 are

mu_1^'=mu=Np
(15)
mu_2^'=Np(1-p+Np)
(16)
mu_3^'=Np(1-3p+3Np+2p^2-3Np^2+N^2p^2)
(17)
mu_4^'=Np(1-7p+7Np+12p^2-18Np^2+6N^2p^2-6p^3+11Np^3-6N^2p^3+N^3p^3),
(18)

so the moments about the mean are

mu_2=Np(1-p)=Npq
(19)
mu_3=Np(1-p)(1-2p)
(20)
mu_4=Np(1-p)[3p^2(2-N)+3p(N-2)+1].
(21)

The skewness and kurtosis excess are

gamma_1=(1-2p)/(sqrt(Np(1-p)))
(22)
=(q-p)/(sqrt(Npq))
(23)
gamma_2=(6p^2-6p+1)/(Np(1-p))
(24)
=(1-6pq)/(Npq).
(25)

The first cumulant is

kappa_1=np,
(26)

and subsequent cumulants are given by the recurrence relation

kappa_(r+1)=pq(dkappa_r)/(dp).
(27)

The mean deviation is given by

MD=sum_(k=0)^N|k-Np|(N; k)p^k(1-p)^(N-k).
(28)

For the special case p=q=1/2, this is equal to

MD=2^(-N)sum_(k=0)^(N)(N; k)|k-1/2N|
(29)
={(N!!)/(2(N-1)!!) for N odd; ((N-1)!!)/(2(N-2)!!) for N even,
(30)

where N!! is a double factorial. For N=1, 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

MD=2(1-p)^(N-|_Np_|)p^(|_Np_|+1)(|_Np_|+1)(N; |_Np_|+1).
(31)

Steinhaus (1999, pp. 25-28) considers the expected number of squares S(n,N,s) containing a given number of grains n on board of size s after random distribution of N of grains,

S(n,N,s)=sP_(1/s)(n|N).
(32)

Taking N=s=64 gives the results summarized in the following table.

nS(n,64,64)
023.3591
123.7299
211.8650
3 3.89221
4 0.942162
5 0.179459
6 0.0280109
7 0.0036840
8 4.16639×10^(-4)
9 4.11495×10^(-5)
10 3.59242×10^(-6)

An approximation to the binomial distribution for large N can be obtained by expanding about the value n^~ where P(n) is a maximum, i.e., where dP/dn=0. Since the logarithm function is monotonic, we can instead choose to expand the logarithm. Let n=n^~+eta, then

ln[P(n)]=ln[P(n^~)]+B_1eta+1/2B_2eta^2+1/(3!)B_3eta^3+...,
(33)

where

B_k=[(d^kln[P(n)])/(dn^k)]_(n=n^~).
(34)

But we are expanding about the maximum, so, by definition,

B_1=[(dln[P(n)])/(dn)]_(n=n^~)=0.
(35)

This also means that B_2 is negative, so we can write B_2=-|B_2|. Now, taking the logarithm of (◇) gives

ln[P(n)]=lnN!-lnn!-ln(N-n)!+nlnp+(N-n)lnq.
(36)

For large n and N-n we can use Stirling's approximation

ln(n!) approx nlnn-n,
(37)

so

(d[ln(n!)])/(dn) approx (lnn+1)-1
(38)
=lnn
(39)
(d[ln(N-n)!])/(dn) approx d/(dn)[(N-n)ln(N-n)-(N-n)]
(40)
=[-ln(N-n)+(N-n)(-1)/(N-n)+1]
(41)
=-ln(N-n),
(42)

and

(dln[P(n)])/(dn) approx -lnn+ln(N-n)+lnp-lnq.
(43)

To find n^~, set this expression to 0 and solve for n,

ln((N-n^~)/(n^~)p/q)=0
(44)
(N-n^~)/(n^~)p/q=1
(45)
(N-n^~)p=n^~q
(46)
n^~(q+p)=n^~=Np,
(47)

since p+q=1. We can now find the terms in the expansion

B_2=[(d^2ln[P(n)])/(dn^2)]_(n=n^~)
(48)
=-1/(n^~)-1/(N-n^~)
(49)
=-1/(Npq)
(50)
=-1/(Np(1-p))
(51)
B_3=[(d^3ln[P(n)])/(dn^3)]_(n=n^~)
(52)
=1/(n^~^2)-1/((N-n^~)^2)
(53)
=(q^2-p^2)/(N^2p^2q^2)
(54)
=(1-2p)/(N^2p^2(1-p)^2)
(55)
B_4=[(d^4ln[P(n)])/(dn^4)]_(n=n^~)
(56)
=-2/(n^~^3)-2/((n-n^~)^3)
(57)
=(2(p^2-pq+q^2))/(N^3p^3q^3)
(58)
=(2(3p^2-3p+1))/(N^3p^3(1-p)^3)).
(59)
BinomialGaussian

Now, treating the distribution as continuous,

lim_(N->infty)sum_(n=0)^NP(n) approx intP(n)dn=int_(-infty)^inftyP(n^~+eta)deta=1.
(60)

Since each term is of order 1/N∼1/sigma^2 smaller than the previous, we can ignore terms higher than B_2, so

P(n)=P(n^~)e^(-|B_2|eta^2/2).
(61)

The probability must be normalized, so

int_(-infty)^inftyP(n^~)e^(-|B_2|eta^2/2)deta=P(n^~)sqrt((2pi)/(|B_2|))=1,
(62)

and

P(n)=sqrt((|B_2|)/(2pi))e^(-|B_2|(n-n^~)^2/2)
(63)
=1/(sqrt(2piNpq))exp[-((n-Np)^2)/(2Npq)].
(64)

Defining sigma^2=Npq,

P(n)=1/(sigmasqrt(2pi))exp[-((n-n^~)^2)/(2sigma^2)],
(65)

which is a normal distribution. The binomial distribution is therefore approximated by a normal distribution for any fixed p (even if p is small) as N is taken to infinity.

If N->infty and p->0 in such a way that Np->lambda, then the binomial distribution converges to the Poisson distribution with mean lambda.

Let x and y be independent binomial random variables characterized by parameters n,p and m,p. The conditional probability of x given that x+y=k is

P(x=i|x+y=k)=(P(x=i,x+y=k))/(P(x+y=k)) =(P(x=i,y=k-i))/(P(x+y=k)) =(P(x=i)P(y=k-i))/(P(x+y=k)) =((n; i)p^i(1-p)^(n-i)(m; k-i)p^(k-i)(1-p)^(m-(k-i)))/((n+m; k)p^k(1-p)^(n+m-k)) =((n; i)(m; k-i))/((n+m; k)).
(66)

Note that this is a hypergeometric distribution.

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