Mixed Lognormal Distribution Calculator

Compute the PDF, CDF (lower probability), upper probability, mean, and variance of a two-component mixed lognormal distribution, with charts.

Input

For the two-component mixed lognormal distribution w・LN(mu1,sigma1)+(1−w)・LN(mu2,sigma2), compute the density, lower probability, and upper probability at x.

Evaluation point x (greater than 0)

Enter the positive value at which to evaluate the probability.

Weight w (between 0 and 1)

Mixing proportion of the first component. The second component uses 1−w.

Component 1: LN(mu1, sigma1)

mu1 (log-mean)

sigma1 (log-std)

Component 2: LN(mu2, sigma2)

mu2 (log-mean)

sigma2 (log-std)

Result

Lower probability F(x) at x = 2

0.69056291

Density f(x)

0.18423052

Upper probability 1 − F(x)

0.30943709

Mean

2.17725762

Variance

6.87909109

Probability density function (PDF)

Cumulative distribution function (CDF)

How it works

The mixed (mixture) lognormal distribution sums two lognormal distributions with weights w and 1−w. Its density is f(x) = w・g(x;mu1,sigma1) + (1−w)・g(x;mu2,sigma2).
Each component density is g(x;mu,sigma) = exp(−(ln x − mu)² / (2 sigma²)) / (x・sigma・sqrt(2 pi)), defined for x greater than 0.
Each component CDF uses the standard normal cumulative distribution: G(x;mu,sigma) = 0.5・erfc(−(ln x − mu) / (sigma sqrt(2))).
The lower probability (CDF) is the weighted sum of the component CDFs, and the upper probability is 1 − CDF.
The mixture mean is the weighted sum of each component mean exp(mu + sigma²/2). The variance comes from the weighted sum of each second moment exp(2 mu + 2 sigma²) minus the squared mean.
Enter a weight w between 0 and 1, and positive values for sigma1 and sigma2.

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