Multiplier Distribution: Histograms and Density Curves

The Probability Density Function (PDF) of multipliers describes the probabilistic structure of possible outcomes of an analytical system. For discrete systems with a finite number of outcomes, the PDF is specified by a probability table: f(xᵢ) = P(X = xᵢ), where Σᵢ f(xᵢ) = 1. For systems with a continuous multiplier space, the PDF is a non-negative function f(x) ≥ 0, satisfying the normalization condition ∫₋∞^∞ f(x)dx = 1. A characteristic feature of the multiplier distribution is its asymmetry: the density is concentrated in the region of small values with a heavy right tail corresponding to rare high multipliers.

The Cumulative Distribution Function (CDF) is defined as F(x) = P(X ≤ x) = ∫₋∞^x f(t)dt and represents a monotonically non-decreasing function with F(−∞) = 0, F(+∞) = 1. CDF is a more robust analysis tool compared to PDF because it is defined for all distributions (including discrete and mixed ones). The quantile function Q(p) = F⁻¹(p) = inf{x : F(x) ≥ p} allows for determining the threshold values of the multiplier for a given probability. For example, the median of the distribution M = Q(0.5), and the 99th percentile Q(0.99) characterizes the upper limit of "typical" values.

The empirical distribution of multipliers is constructed based on observed data using histograms and Kernel Density Estimation (KDE). The histogram divides the range of values into k intervals (bins) of equal width h = (x_max − x_min)/k, and the relative frequency is calculated for each bin. The optimal number of bins is determined by Sturges’ rule (k = 1 + log₂n), Freedman-Diaconis’ rule (h = 2·IQR·n⁻¹/³), or Scott’s rule (h = 3.49·s·n⁻¹/³). KDE provides a continuous density estimate: f̂(x) = (1/n·h)·Σᵢ K((x − xᵢ)/h), where K is a kernel function (typically a Gaussian kernel).

The estimation of theoretical distribution parameters that best approximate empirical data is carried out by the maximum likelihood estimation (MLE) method. The likelihood function L(θ) = Πᵢ f(xᵢ|θ) is maximized over the parameter vector θ. For multiplier distributions, the following are often applied: exponential distribution (f(x) = λe⁻λˣ), gamma distribution (f(x) = xᵅ⁻¹·e⁻ˣ/β / (βᵅ·Γ(α))), and Pareto distribution (f(x) = α·x_m^α / x^(α+1)). The choice of the optimal model is made through information criteria AIC = 2k − 2ln(L) and BIC = k·ln(n) − 2ln(L), as well as goodness-of-fit tests (Pearson’s χ²-test, Kolmogorov-Smirnov test, Anderson-Darling test).