Binomial Distribution

The Binomial Distribution is the discrete probability distribution of the number of successes in a sequence of n independent yes/no experiments, each of which yields success with probability p. Such a success/failure experiment is also called a Bernoulli experiment or Bernoulli trial; when n = 1, the Binomial Distribution is a Bernoulli Distribution.

Assumptions

n is a strictly positive Integer and p is a valid probability (0 ≤ p ≤ 1).

Use

mctad.binomial(n, p)

Inline Comments

mctad.binomial = function (n, p) {

Check that p is a valid probability (0 ≤ p ≤ 1), and that n is an integer, strictly positive.

if (p < 0 || p > 1.0 || !mctad.isInteger(n) || n <= 0) { return undefined; } var x = 0, pmf, cdf = 0, dfs = { mean: n * p, median: undefined, mode: function () { if ((n + 1) * p === 0.0 || !mctad.isInteger((n + 1) * p)) { return [Math.floor((n + 1) * p)]; } else { if (mctad.isInteger((n + 1) * p) && (n + 1) * p >= 1 && (n + 1) * p <= n) { return [(n + 1) * p - 1, (n + 1) * p]; } else { return n; } } }(), variance: (n * p) * (1.0 - p), skewness: (1 - 2 * p)/Math.sqrt(n * p * (1.0 - p)), entropy: undefined, // @todo: implement from wikipedia once O(1/n) becomes clear domain: { min: 0, max: Infinity }, range: { min: 0.0, max: 0.0 },

mctad.binomial(9, .7).generate(100) will perform 100 sequences of nine Bernoulli trials where the random variables have a success probability of .7.

generate: function (m) { var trial = [], randomVariables = []; for (var i = 0; i < m; i++ ) { trial = mctad.bernoulli(p).generate(n); randomVariables.push(mctad.sum(trial)); } return randomVariables; } };

Iterate over the domain, calculating the probability mass and cumulative distribution functions.

do { pmf = (mctad.factorial(n) / (mctad.factorial(x) * mctad.factorial(n - x)) * (Math.pow(p, x) * Math.pow(1.0 - p, (n - x)))); cdf += pmf; dfs[x] = { pmf: pmf, cdf: cdf }; if (pmf > dfs.range.max) { dfs.range.max = 0.1 * Math.ceil(10 * pmf); } x++; } while (dfs[x - 1].cdf < 1.0 - mctad.ε); dfs.domain.max = x - 1;

Mix in the convenience methods for p(x) and F(x).

mctad.extend(dfs, mctad.discreteMixins); return dfs; };