Mathematics

Binomial distribution

Bernoulli experiment, probability function, expected value, standard deviation and normal approximation using Moivre-Laplace and Yates's correction.

An experiment follows the model of the binomial or Bernoulli distribution if:
  • In each trial of the experiment only two outcomes are possible: the event A (success) and its opposite Aˉ\bar{A} (failure).
  • The probability of event A occurring is constant, represented by p\mathbf{p}.
  • The result obtained in each trial is independent of the results obtained previously.
The binomial distribution is usually represented by B(n,p)\mathbf{B(n, p)}, where n\mathbf{n} is the number of trials, p\mathbf{p} is the probability of success, and q\mathbf{q} is the probability of failure, being q=1−pq = 1 - p.
The probability of obtaining k\mathbf{k} successes in a binomial is:
P(x=k)=(nk)pk⋅qn−kP(x = k) = \binom{n}{k} p^k \cdot q^{n-k}
Expected value or mean (μ\mu): It is a measure of central tendency that is used to designate a collection of elements by a single value and is represented by μ\mu. It is given by:
μ=n⋅p\mu = n \cdot p
Standard deviation (σ\sigma): It is a measure of dispersion that is used to indicate how close the elements of the collection are to the mean and is represented by σ\sigma. In a binomial distribution it is given by:
σ=n⋅p⋅q\sigma = \sqrt{n \cdot p \cdot q}
Moivre-Laplace Theorem:
If X is a discrete variable that follows a binomial distribution with parameters n and p, B(n,p)B(n, p), and it is true that n>10n > 10, n⋅p>5n \cdot p > 5 and n⋅q>5n \cdot q > 5, it is a fairly good approximation to assume that the variable X approximates the normal variable X′X' with mean μ=n⋅p\mu = n \cdot p and standard deviation σ=n⋅p⋅q\sigma = \sqrt{n \cdot p \cdot q}, that is:
B(n,p)  ⟹  N(n⋅p,n⋅p⋅q)B(n, p) \implies N(n \cdot p, \sqrt{n \cdot p \cdot q})
To calculate probabilities with the approximation to the Normal we must take into account that the binomial is discrete and the normal is continuous, so we introduce an adjustment in the calculation called Yates's Correction:
P(X=a)=P(a−0.5≤X′≤a+0.5)P(X≤a)=P(X′≤a+0.5)P(X≥a)=P(X′≥a−0.5)P(X<a)=P(X′≤a−0.5)P(X>a)=P(X′≥a+0.5)\begin{aligned} P(X = a) &= P(a - 0.5 \le X' \le a + 0.5) \\ P(X \le a) &= P(X' \le a + 0.5) \quad & P(X \ge a) &= P(X' \ge a - 0.5) \\ P(X < a) &= P(X' \le a - 0.5) \quad & P(X > a) &= P(X' \ge a + 0.5) \end{aligned}