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    Logistic Distribution

    A key distribution in machine learning and growth modeling.

    The "Growth Curve" Distribution

    The Logistic distribution is a continuous probability distribution whose cumulative distribution function is the logistic function, which appears in logistic regression and feedforward neural networks. It resembles the normal distribution but has heavier tails, meaning it gives more probability to extreme events.

    In finance, it's used in credit risk modeling to estimate the probability of default. Its S-shaped cumulative distribution function is perfect for modeling phenomena that have a "saturation" point, like the adoption rate of a new technology or the market share of a product.

    The Formula
    The probability density function (PDF) is given by:
    f(x;μ,s)=e−(x−μ)/ss(1+e−(x−μ)/s)2f(x; \mu, s) = \frac{e^{-(x-\mu)/s}}{s(1+e^{-(x-\mu)/s})^2}f(x;μ,s)=s(1+e−(x−μ)/s)2e−(x−μ)/s​
    • μ\muμ (mu) is the location parameter, which is also the mean, median, and mode.
    • s>0s > 0s>0 is the scale parameter, which is proportional to the standard deviation. A larger scale value makes the curve wider and flatter.
    Interactive Logistic Distribution
    Adjust the location (μ) and scale (s) parameters to see how the shape of the distribution changes.