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Herschel-Maxwell derivation of the normal distribution

zeitbach.com · 2,246 words · saved by 1 readers

The probability density function (PDF) for the normal distribution is given by the following (somewhat intimidating) formula: 𝑓 ( 𝑥 ) = 1 𝜎 2 𝜋 𝑒 − 1 2 ( 𝑥 − 𝜇 𝜎 ) 2 It describes the well-known bell-shaped curve that is centered around the mean 𝜇 , has inflection points at one standard deviation 𝜎 around the mean, and integrates to one. Because of the central limit theorem, this curve is lurking behind every corner in statistics. I recently read three beautiful derivations of this function in Jaynes (2003). Here I’m expanding on the Herschel-Maxwell derivation, following the steps of this video. I’m starting off with a thought experiment given by Herschel himself in 1869: Suppose a ball dropped from a given height, with the intention that it shall fall on a given mark.1 Imagine we repated this little experiment a thousand times. We might end up with something like this: Each dot represents a landing spot of the dropped ball. I drew a Cartesian coordinate system over the do

Herschel-Maxwell derivation of the normal distribution zeitbach.com The probability density function (PDF) for the normal distribution is given by the following (somewhat intimidating) formula: f ( x ) = 1 σ 2 π e − 1 2 ( x − μ σ ) 2 f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{1}{2} \lparen \frac{x - \mu}{\sigma} \rparen^2} It describes the well-known bell-shaped curve that is centered around the mean μ \mu , has inflection points at one standard deviation σ \sigma around the mean, and integrates to one. Because of the central limit theorem , this curve is lurking behind every corner in stat

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