Logistic Distribution

The logistic distribution, also known as the sech-squared distribution, is a continuous probability distribution used to model growth. It is characterized by its S-shaped cumulative distribution function (CDF), which is similar to the normal distribution but with heavier tails.

The logistic distribution has a location parameter $a$ and a scale parameter $b$. The probability density function (PDF) is given by:

The cumulative distribution function (CDF) is:

The logistic distribution spans the entire real number line from negative to positive infinity. Its scale parameter b must be strictly positive.

• Growth modeling in population dynamics and ecology uses the logistic distribution to model S-shaped growth curves.

• Machine learning applications employ logistic distributions in neural networks and logistic regression.

• Marketing research uses the distribution to model adoption rates of new products.

• Survival analysis uses logistic distributions in lifetime modeling.

Notable properties include:

• The characteristic function is $\varphi (t)=e^{iat}\pi bt/\sinh (\pi bt)$.

• The quantile function is $Q(p)=a+b\ln (p/(1-p))$.

• The distribution is symmetric around its mean.

• The logistic distribution is a special case of the generalized logistic distribution.

• The difference between two independent Gumbel random variables follows a logistic distribution.

• The distribution approaches the normal distribution when standardized and properly scaled.

The logistic distribution is implemented by the LogisticDistribution class. It has one constructor that takes two arguments. The first argument is the location parameter, and corresponds to the mode of the probability density function. The second argument is the scale parameter.

The following constructs the same logistic distribution with location parameter 6.8 and scale parameter 4.1:

var logistic = new LogisticDistribution(6.8, 4.1);

Dim logistic = New LogisticDistribution(6.8, 4.1)

No code example is currently available or this language may not be supported.

let logistic = LogisticDistribution(6.8, 4.1)

The LogisticDistribution class has two specific properties, LocationParameter and ScaleParameter, which return the location parameter (mode) and scale parameter of the distribution.

LogisticDistribution has one static (Shared in Visual Basic) method, Sample, which generates a random sample using a user-supplied uniform random number generator. The second and third parameters are the location and scale parameters of the distribution.

var random = new Pcg32();
double sample = LogisticDistribution.Sample(random, 6.8, 4.1);

Dim random = New Pcg32()
Dim sample = LogisticDistribution.Sample(random, 6.8, 4.1)

No code example is currently available or this language may not be supported.

let random = Pcg32()
let sample = LogisticDistribution.Sample(random, 6.8, 4.1)

The above example uses the MersenneTwister class to generate uniform random numbers.

For more information on the logistic distribution, refer to the following sources:

• Johnson, N. L., Kotz, S., & Balakrishnan, N. (1994). "Continuous Univariate Distributions, Volume 2", Chapter 23. Wiley Series in Probability and Statistics.

• Forbes, C., Evans, M., Hastings, N., & Peacock, B. (2011). "Statistical Distributions", Chapter 28. Wiley Series in Probability and Statistics.

For details of the properties and methods common to all continuous distribution classes, see the topic on continuous distributions.

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