Distributions, Expectation & Variance

A distribution answers: over all possible outcomes, how is belief spread? That spread has a shape, and the shape is everything:

• Uniform: every outcome equally likely — maximum shrugging, like a fair die
• Normal (bell curve): mass piles at the center with rare extremes — heights, measurement noise, and anything that's a sum of many small effects
• Long-tailed: rare events aren't that rare — wealth, word frequencies, city sizes. 'Once-in-a-century' events arrive yearly

ML depends on distributions everywhere: your training data is one, and the model quietly learns its shape.