Pascal Distribution

The Pascal or Negative Binomial Distribution is a discrete probability distribution of the number of successes in a sequence of independent and identically distributed Bernoulli trials before a specified number of failures, r, occurs. p is the probability of success. This implementation restricts r to be integer-valued.

Assumptions

r is a strictly positive Integer, p is a valid probability (0 < p < 1).

Use

mctad.pascal(r, p)

Inline Comments

mctad.pascal = function (r, p) {

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

if (p <= 0 || p >= 1.0 || !mctad.isInteger(r) || r <= 0) { return undefined; } var k = 0, pmf, cdf = 0, dfs = { mean: (r * p) / (1.0 - p), median: undefined, mode: (function () { if (r > 1) { return [Math.floor((p * (r - 1)) / (1.0 - p))]; } else { return [0]; } })(), variance: (r * p) / Math.pow((1.0 - p), 2), skewness: (1 + p) / Math.sqrt(r * p), entropy: undefined, domain: { min: 0, max: Infinity }, range: { min: 0.0, max: 0.0 },

mctad.pascal(5, 0.8).generate(100) will perform 100 sequences of Bernoulli trials, each of which ceases when five failures have been observed. The probability of success is 0.8.

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

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

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

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

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