Mixed Lognormal Percent Point Calculator

Find the percent point (quantile) x for a probability p in a two-component mixed lognormal distribution. Set the weight, each component mu and sigma, and the mixing mode.

Input

Enter probability p, the mixing mode, weight w, and the mu and sigma of each component to invert the mixed lognormal CDF and find the percent point x.

Probability p

Cumulative probability greater than 0 and less than 1

Mixing mode

Choose how the components are combined

Weight w (component 1)

Weight of component 1. Component 2 gets 1-w.

Component 1 (lognormal)

mu1 (mean of log)

sigma1 (std dev of log)

Component 2 (lognormal)

mu2 (mean of log)

sigma2 (std dev of log)

Result

Percent point x at probability p = 0.9

4.68055942

F(x) (check)

0.9

f(x) (density)

0.03470908

Component 1 F(x)

0.9989885

Component 2 F(x)

0.75151725

Component 1 median

Component 2 median

2.71828183

Probability density f(x)

Cumulative distribution F(x)

How it works

A mixed lognormal distribution blends two lognormal distributions LN(mu1,sigma1) and LN(mu2,sigma2) with weights w and 1-w.
Each component CDF is F_i(x)=Phi((ln x - mu_i)/sigma_i), where Phi is the standard normal CDF.
For the cdf or density mode, the mixed CDF is F(x)=w F_1(x)+(1-w)F_2(x). Density mixing and CDF mixing give the same percent point.
For the max mode, the distribution is the independent maximum of the two components, so F(x)=F_1(x) times F_2(x). The weight is ignored.
For the min mode, the distribution is the independent minimum, so F(x)=1-(1-F_1(x))(1-F_2(x)). The weight is ignored.
The percent point is found by inverting F(x)=p with bisection. The probability p must be greater than 0 and less than 1.
The standard normal CDF is computed with the erf function, combining a series expansion and a continued fraction for high accuracy.

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