Lognormal Distribution

The Lognormal Distribution is a continuous probability distribution of a random variable whose logarithm is normally distributed.

Assumptions

μ and σ2 are real numbers, the mean and variance.

Use

mctad.lognormal(μ, σ2)

Inline Comments

mctad.lognormal = function (μ, σ2) {

Check that μ > 0 and σ2 > 0.

if (μ < 0 || σ2 <= 0) { return undefined; } var dfs = { mean: Math.pow(Math.E, μ + σ2 / 2), median: Math.pow(Math.E, μ), mode: Math.pow(Math.E, μ - σ2), variance: Math.pow(Math.E, 2 * μ + σ2) * (Math.pow(Math.E, σ2) - 1), skewness: (Math.pow(Math.E, σ2) + 2) * Math.sqrt(Math.pow(Math.E, σ2) - 1), entropy: 0.5 + 0.5 * Math.log(2 * mctad.π * σ2) + μ, domain: { min: 0.0, max: Infinity }, range: { min: 0, max: Infinity },

mctad.lognormal(2.0, 0.5).generate(100) will generate an Array of 100 random variables, distributed lognormally with mean 2 and variance 0.5.

generate: function (n) { var randomVariables = []; randomVariables = mctad.normal(μ, σ2).generate(n); for (var i = 0; i < n; i++ ) { randomVariables[i] = Math.pow(Math.E, randomVariables[i]); } return randomVariables; }, pdf: function (x) { if (x > 0) { return (1 / (x * Math.sqrt(2 * mctad.π * σ2))) * Math.pow(Math.E, -(Math.pow(Math.log(x) - μ, 2) / (2 * σ2))); } else { return undefined; } }, cdf: function (x) { if (x > 0) { var Z = (Math.log(x) - μ) / Math.sqrt(σ2); return mctad.normal(0, 1).F(Z); } else { return 0.0; } } };

Mix in the convenience methods for f(X) and F(X).

mctad.extend(dfs, mctad.continuousMixins); dfs.domain.max = μ + Math.ceil(2.5 * dfs.variance); dfs.range.max = 0.1 * Math.ceil(10 * dfs.pdf(dfs.mode)); return dfs; };