Probability Distributions Explained - Gaussian, Bernoulli and Von Mises

Two wind readings, 350 degrees and 10 degrees, both say the wind is from the north. Their average is 180 degrees: due south. What went wrong? The standard probability distributions behind neural network outputs, each built from one weather station and proved on screen: the Bernoulli and binomial (and why maximum likelihood can declare rain impossible), the multinomial with a Lagrange-multiplier derivation, the multivariate Gaussian (ellipses from eigenvectors, conditioning by completing the square, marginals, sequential estimation), mixtures of Gaussians, the von Mises distribution for directions, the exponential family (where the logistic sigmoid comes from, and sufficient statistics), and nonparametric density estimation with histograms, kernels and nearest neighbours. Chapters 0:00 Why the average wind direction points the wrong way 0:26 Bernoulli, binomial and multinomial: rain and sky 2:05 The multivariate Gaussian: why its contours are ellipses 3:27 Conditioning and marginals: forecasting tomorrow from today 4:41 Mixtures of Gaussians 5:11 Directions: arrows, not angles (von Mises) 6:06 The exponential family and the logistic sigmoid 6:58 Histograms, kernel densities and nearest neighbours 7:46 Recap and three questions for you Questions to think about (answer in the comments): 1. After three dry days, maximum likelihood says rain is impossible. How could a prior on mu fix that? 2. If today and tomorrow were uncorrelated, how much would knowing today shrink the spread for tomorrow? 3. What is the natural parameter of the multinomial, and which neural network output do you get by inverting it? Subtitles: English, Hindi. Part of the playlist "Deep Learning Basics". Further reading: C. M. Bishop and H. Bishop, Deep Learning: Foundations and Concepts (Springer, 2024), chapter 3. This video contains AI-generated content: the narration is AI-synthesised and the visuals are generated programmatically. © 2026 The Deep Learn. All rights reserved. #DeepLearning #MachineLearning #Probability