Euler's gamma function, introduced in the 18th century, is one of the cornerstones of modern probability distribution theory, with fundamental applications in contemporary artificial intelligence. But how exactly does this mathematical function regulate data complexity and guide machine learning? Euler's gamma function, defined as $\Gamma(z) = \int_0^+\infty t^z-1 e^-t dt$ for $z > 0$, generalises the factorial and proves essential for describing continuous distributions that model real phenomena. Unlike the Poisson or classical Gamma distribution, the gamma acts as a flexible parameter that regulates the shape, scale and support of random variables, allowing for a more accurate description of variability in data. Among the distributions that include it, the Gamma, Beta and Chi-square distributions stand out, all of which are used in statistics and machine learning.In the context of AI, the gamma function plays a crucial role in model training. When working with neural networks or probabilistic models, one often encounters continuous variables that require flexible representation: gamma acts as a scale parameter that adapts the dispersion and asymmetry of distributions. For example, in the multivariate Gamma distribution, $\Gamma(\alpha_1, \alpha_2, \dots, \alpha_k)$ governs the shape of the density, allowing complex relationships between inputs and outputs to be captured in regression or classification algorithms. This enables better estimation of uncertainties and more reliable predictions.In deep learning, the gamma function is used in advanced sampling techniques, such as Gaussian Process Regression or Variational Autoencoder (VAE) models, where synthetic data is generated following distributions governed by gamma parameters. Furthermore, when calculating the likelihood function, the gamma function allows sampled distributions to be normalised, ensuring that the probabilities add up to 1. A concrete example can be found in image datasets: the gamma distribution models pixel brightness in a non-linear way, improving the robustness of recognition in low-light conditions. The gamma function is not just a mathematical artifice: it is the pillar that makes key distributions in AI operational. Consider the Beta distribution, used to model probabilities in Bayesian models: it contains a gamma parameter that defines its concentration. Similarly, in a neural network that predicts probabilities, gamma scaling the outputs ensures that the outputs remain in $[0,1]$ and comply with physical or statistical constraints. The Chi-square distribution, derived from the sum of independent gamma variables, is fundamental in model validation tests, where the goodness of fit is evaluated.While the exponential distribution $\lambda e^-\lambda x$ describes waiting times, and the log-normal models highly skewed variables, the gamma combines flexibility and precision: it is the most versatile distribution for positive and skewed variables. In Italian contexts, such as the analysis of health or financial data, the gamma is often preferred for modelling recovery times, cumulative expenses or traffic volumes, where the presence of zero or extreme values requires a flexible but controlled distribution.Euler's legacy, which originated in Europe, lives on today in Italy through the spread of open source libraries such as SciPy and PyTorch, which implement algorithms based on the gamma function. Italian universities, such as the Politecnico di Milano and the University of Padua, are conducting research on advanced probabilistic models in which the gamma function regulates the shape of distributions. This combination of mathematical tradition and technological innovation is an invisible but powerful driver of the evolution of artificial intelligence in our country. The gamma function is not just an abstract concept: it is the connective tissue that links theory and practice in modern AI. Understanding its role in distributions allows us to better interpret data, improve models and manage uncertainty with greater awareness. In the coming era of machine learning, where data is complex and multivariate, the gamma continues to drive innovation, demonstrating that pure mathematics is the foundation of future technology. Table of contents “The gamma function is not just an integral: it is the thread that links elegant mathematics to real-world applications, guiding artificial intelligence towards a deeper understanding of data.”For more information, see the original topic: Euler's gamma function and its role in modern probability distributions.