Bernoulli Distribution

The Bernoulli distribution is the probability discrete distribution of a random variable which takes value 1 with success probability p and value 0 with failure probability q = 1 - p. It can be used, for example, to represent the toss of a coin, where "1" is defined to mean "heads" and "0" is defined to mean "tails" (or vice versa). It is a special case of the Binomial Distribution where n = 1.

Assumptions

p is a valid probability (0 ≤ p ≤ 1).

Use

mctad.bernoulli(p)

Inline Comments

mctad.bernoulli = function(p) {

Check that p is a valid probability (0 ≤ p ≤ 1)

if (p < 0 || p > 1.0) { return undefined; } var x, dfs = { mean: p, median: (function () { if (p < 0.5) { return 0.0; } else { if (p === 0.5) { return 0.5; } else { return 1.0; } } })(), mode: (function () { if ((p < 0.5)) { return [0.0]; } else { if (p === 0.5) { return [0, 1]; } else { return [1.0]; } } })(), variance: p * (1.0 - p), skewness: ((1.0 - p) - p)/Math.sqrt(p * (1.0 - p)), entropy: -(1.0 - p) * Math.log(1.0 - p) - p * Math.log(p), domain: { min: 0, max: 1 }, range: { min: 0.0, max: 0.0 },

mctad.bernoulli(.7).generate(100) will perform 100 Bernoulli trials, yielding 100 random variables each having had a success probability of .7.

generate: function (n) { var randomVariables = []; for (var i = 0; i < n; i++ ) { if (mctad.getRandomArbitrary(0, 1) <= p) { randomVariables.push(1); } else { randomVariables.push(0); } } return randomVariables; } };

Assign the probability mass and cumulative distribution functions for the outcomes 0 or 1.

dfs[0] = { pmf: 1.0 - p, cdf: 1.0 - p }; dfs[1] = { pmf: p, cdf: 1.0 }; if (p > 1.0 - p) { dfs.range.max = 0.1 * Math.ceil(10 * p); } else { dfs.range.max = 0.1 * Math.ceil(10 * (1.0 - p)); }

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

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