Poisson Distribution

The Poisson Distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time and/or space if these events occur with a known average rate and independently of the time since the last event.

The Poisson Distribution is characterized by the strictly positive mean arrival or occurrence rate, λ.

Assumptions

λ is a positive Real number.

Use

mctad.poisson(λ)

Inline Comments

mctad.poisson = function (λ) {

Check that λ is strictly positive.

if (λ <= 0) { return undefined; } var x = 0, pmf, cdf = 0, dfs = { mean: λ, median: Math.floor(λ + 1 / 3 - 0.02 / λ), mode: [Math.ceil(λ) - 1, Math.floor(λ)], variance: λ, skewness: Math.pow(λ, 0.5), entropy: undefined, // @todo: revisit this domain: { min: 0, max: Infinity }, range: { min: 0.0, max: 0.0 },

mctad.poisson(10).generate(100) will generate an Array of 100 random variables, having a Poisson Distribution with an arrival rate of 10 per time unit.

generate: function (n) { var a = Math.pow(Math.E, -λ), randomVariables = []; for (var i = 0; i < n; i++) { var j = 0, b = 1; do { b = b * mctad.getRandomArbitrary(0, 1); j++; } while (b > a); randomVariables.push(j - 1); } return randomVariables; } };

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

do { pmf = (Math.pow(Math.E, -λ) * Math.pow(λ, x)) / mctad.factorial(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; };