Dot product: Component vs. Geometric definition

The Geometry of Uncertainty: A Fresh Perspective on Probability Distributions

A geometric and combinatorial reconstruction of the most fundamental probability distributions — exponential, Poisson, and normal — building from first principles rather than formulas. The exponential distribution is presented as the most primitive (requiring only a constant rate), Poisson as its counting consequence, and the normal distribution as the inevitable attractor of repeated convolution, visibly hiding inside Pascal's Triangle. The Law of Large Numbers is derived from just three simple properties (linearity of sum, variance scaling, mean scaling), revealing that the sample mean is an entirely different random variable from the original — one that converges toward certainty while individual outcomes remain permanently uncertain. The core insight is that independent deviations cannot conspire, making variance grow linearly rather than quadratically, and that this sub-quadratic growth is the true engine of the LLN.