Research VIII) Binomial distribution and its relatives

Research VIII) Binomial distribution and its relatives

In probability, the binomial distribution with parameters n and p is the discrete probability distribution of the number of successes in a sequence of n independent experiments, success (with probability p) or failure (with probability q = 1 − p). A single success/failure experiment is also called a Bernoulli trial, so for a single trial, i.e., n = 1, the binomial distribution is a Bernoulli distribution.

In general, if the random variable X follows the binomial distribution with parameters n ∈ ℕ and p ∈ [0,1], we write X ~ B(n, p). The probability of getting exactly k successes in n trials is given by the probability mass function:

{\displaystyle Pr(k;n,p)=\Pr(X=k)={n \choose k}p^{k}(1-p)^{n-k}}

for k = 0, 1, 2, …, n
where {\binom {n}{k}}={\frac {n!}{k!(n-k)!}} is the so called binomial coefficent. A binomial coefficient is indexed by a pair of integers n ≥ k ≥ 0 and is written like{\displaystyle {\tbinom {n}{k}}.} It is the coefficient of the xk term in the polynomial expansion of the binomial power (1 + x)n, and it is given by the formula shown above.

The binomial coefficients occur in many areas of mathematics, especially in the field of combinatorics. {\tbinom {n}{k}} is often read aloud as “n choose k“, because there are {\tbinom {n}{k}} ways to choose a subset of size k elements, disregarding their order, from a set of n elements. The properties of binomial coefficients have led to extending the definition to beyond the common case of integers n ≥ k ≥ 0.

The formula can be understood as follows: k successes occur with probability pk and n − k failures occur with probability (1 − p)n − k. However, there can be permutations of the trials, such that the k successes can occur anywhere among the n trials; in other words, there are{n \choose k}different ways of distributing k successes in a sequence of n trials.

 

bin
a summary of the Binomial distribution characteristics

If X ~ B(n, p), that is, X is a binomially distributed random variable, n being the total number of experiments and p the probability of each experiment yielding a successful result, then the expected value of X is:

{\displaystyle \operatorname {E} [X]=np.}

For example, if n = 100, and p =1/4, then the average number of successful results will be 25.

Proof: We calculate the mean, μ, directly calculated from its definition.

{\displaystyle \mu =\sum _{i=0}^{n}x_{i}p_{i},} 

so, given the definition the proof is as it follows:
bern

 

This is possible thanks to the Binomial theorem. According to the theorem, it is possible to expand the polynomial (x + y)n into a sum involving terms of the forma xb yc, where the exponents b and c are nonnegative integers with b + c = n, and the coefficient a of each term is a specific positive integer depending on n and b. For example,

{\displaystyle (x+y)^{4}=x^{4}+4x^{3}y+6x^{2}y^{2}+4xy^{3}+y^{4}.}

The coefficient a in the term of a xb yc is known as the binomial coefficient {\tbinom {n}{b}} or {\tbinom {n}{c}} (the two have the same value). These numbers also arise in combinatorics, where {\tbinom {n}{b}} gives the number of different combinations of b elements that can be chosen from an n-element set.

It is also possible to deduce the mean from the equation {\displaystyle X=X_{1}+\cdots +X_{n}} whereby all X_{i} are Bernoulli distributed random variables with E[X_{i}]=p.
We get{\displaystyle E[X]=E[X_{1}+\cdots +X_{n}]=E[X_{1}]+\cdots +E[X_{n}]=\underbrace {p+\cdots +p} _{n{\text{ times}}}=np}

As well as with the mean, it will be easy to show how the Binomial Variance can be calculated. The variance is:

{\displaystyle \operatorname {Var} (X)=np(1-p).}

Proof:
Let {\displaystyle X=X_{1}+\cdots +X_{n}} where all X_{i} are independently Bernoulli distributed random variables. Since {\displaystyle \operatorname {Var} (X_{i})=p(1-p)}, we get:

{\displaystyle \operatorname {Var} (X)=\operatorname {Var} (X_{1}+\cdots +X_{n})=\operatorname {Var} (X_{1})+\cdots +\operatorname {Var} (X_{n})=n\operatorname {Var} (X_{1})=np(1-p).}

Gven the Binomial distribution, there exist different derivations, which is interesting to talk about:

  • Sums of binomials

    If X ~ B(n, p) and Y ~ B(m, p) are independent binomial variables with the same probability p, then X + Y is again a binomial variable; its distribution is Z=X+Y ~ B(n+m, p):{\begin{aligned}\operatorname {P} (Z=k)&=\sum _{i=0}^{k}\left[{\binom {n}{i}}p^{i}(1-p)^{n-i}\right]\left[{\binom {m}{k-i}}p^{k-i}(1-p)^{m-k+i}\right]\\&={\binom {n+m}{k}}p^{k}(1-p)^{n+m-k}\end{aligned}}

    However, if X and Y do not have the same probability p, then the variance of the sum will be smaller than the variance of a binomial variable distributed asB(n+m,{\bar {p}}).\,

  • Bernoulli distribution

    As we said before, the Bernoulli distribution is a special case of the binomial distribution, where n = 1. Symbolically, X ~ B(1, p) has the same meaning as X ~ B(p). Conversely, any binomial distribution, B(n, p), is the distribution of the sum of n Bernoulli trials, B(p), each with the same probability p.

  • Poisson binomial distribution

    The binomial distribution is a special case of the Poisson binomial distribution, or general binomial distribution, which is the distribution of a sum of n independent non-identical Bernoulli trials B(pi).

  • Normal approximation

    Binomial probability mass function and normal probability density function approximation for n = 6 and p = 0.5

    If n is large enough, then the skew of the distribution is not too great. In this case a reasonable approximation to B(n, p) is given by the normal distribution

    {\displaystyle {\mathcal {N}}(np,\,{\sqrt {np(1-p)}}),}

    and this basic approximation can be improved using a suitable continuity correction. The basic approximation generally improves as n increases (at least 20) and is better when p is not near to 0 or 1.

     

 

An interesting aspect as for the Binomial derivations are the Limiting distributions:

  • Poisson limit theorem: As n approaches ∞ and p approaches 0, then the Binomial(n, p) distribution approaches the Poisson distribution with expected value λ = np.
  • De Moivre–Laplace theorem: As n approaches ∞ while p remains fixed,
{\frac {X-np}{\sqrt {np(1-p)}}}
the distribution of approaches the normal distribution with expected value 0 and variance 1.
This result is sometimes loosely stated by saying that the distribution of X is asymptotically normal with expected value np and variance np(1 − p).
This result is a specific case of the central limit theorem, about which we’ll talk in the next research.

 

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