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.

Nguồn: https://medium.com/@tomkob99_89317/the-geometry-of-uncertainty-a-fresh-perspective-on-probability-distributions-e43ec051b642. 8sync News chỉ tóm tắt và dẫn link; bản quyền nội dung thuộc tác giả và nguồn gốc.

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What To Learn To Be A Graphics Programmer

A practical guide for aspiring graphics programmers covering what to learn to become hireable. Graphics programming is framed as two parallel tracks: the CPU side (explicit APIs like DirectX 12, Vulkan, Metal) and the GPU side (lighting math, shading, shadows, ambient occlusion, post-processing). Recommended learning resources include 'Ray Tracing in One Weekend' for path tracing, learnopengl.com for PBR theory, the Filament documentation for deeper PBR, and the PBRT book. Required math is manageable — linear algebra, basic trigonometry, and some calculus — while knowledge of standard data structures and algorithms is also advised. The author briefly notes skepticism about LLM hype but acknowledges ML has a place in the toolbox.

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