Normal Distribution

Math for CS · Axiom Academy

The bell curve, Z-scores, and the Central Limit Theorem The Most Important Distribution The normal (or Gaussian ) distribution is the most important distribution in probability and statistics. Its bell-curve shape appears everywhere: test scores, measurement errors, heights, stock returns, network latencies, and countless other phenomena. Parameters: (mean, the center) and (variance, the spread). The PDF is a symmetric bell curve centered at . The standard normal has mean 0 and variance 1. Any normal variable can be converted to standard form: This is called standardization or computing the Z-score . The Z-score tells you how many standard deviations a value is from the mean. For any normal distribution, the probability within k standard deviations of the mean is: Computing Normal Probabilities For , convert to the standard normal and look up : The most profound result in probability theory: In plain English: the average of many independent random variables is approximately normal, regardless of the original distribution. Total load across many servers is the sum of many independent contributions. By CLT, the total is approximately normal, letting us predict overload probability. Many ML algorithms assume features are normally distributed. Gaussian Naive Bayes, linear discriminant analysis, and regularization techniques all rely on normality assumptions.

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