Why the Normal Distribution Has the Shape e^(-x²): Deriving the Gaussian from First Principles

Friends, thank you all for your comments. Every time I read your feedback, I realize — the audience of this channel is something special. You’re not just studying mathematics, you’re noticing genuine mathematical beauty.

We’ll eventually cover everything that’s been mentioned, but as promised, let’s start with the most upvoted topic.

Five years ago, around this time, I published one of my first videos on the Law of Large Numbers — the idea that the observed frequency of events converges to their theoretical probability. It’s a simple and remarkable statement, and it’s not immediately clear whether it’s a law of nature or a mathematical theorem. Yet this law governs an enormous number of things in our world. For the first two years, the channel grew almost entirely thanks to that one video.

Right after the Law of Large Numbers came a video on the Normal Distribution — the bell-shaped curve that shows up everywhere. But a very natural question arises:

Why Does the Normal Distribution Have the Shape e^(-x²)?

There are countless functions that look like a hill with slopes. So why is it that when I throw snowballs at a post, the spread traces out exactly this particular function?

The answer is actually straightforward: the shape of the distribution emerges from three simple properties of random errors — symmetry, independence, and the rarity of large deviations.

Three Assumptions Behind the Normal Distribution Derivation

1. Symmetry — the probability of an error does not depend on its direction. A miss to the left and a miss upward are equally likely.

2. Independence — horizontal and vertical errors are independent: missing to the left has no effect on the probability of missing upward.

3. Small errors are more common — large deviations from the center are less probable than small ones.

Deriving the Gaussian Formula Step by Step

Let p(x) be the probability of an error of magnitude x in the horizontal direction, and p(y) the same for the vertical direction.

From independence it follows that: p(x) · p(y) = g(r), where r = x² + y² is the distance from the center.

Since the result must be independent of coordinate rotation (symmetry), we arrive at the equation:

p′(x) / (x · p(x)) = p′(y) / (y · p(y)) = C, where C is a constant.

Solving this equation gives: p(x) = A · e^(C · x² / 2).

But by our third assumption (small errors are more likely than large ones), C must be negative. Writing C = −2k², where k > 0, we get:

p(x) = A · e^(−k² · x²)

where A is the normalization coefficient and k determines the width of the distribution.

In other words, the shape e^(−x²) is the only shape that preserves both symmetry and independence of errors under any rotation of coordinates.

This is the fundamental form of the Gaussian.

Why the Gaussian Curve Appears Everywhere in Nature

If errors are independent, symmetric, and large deviations are rare — the only possible distribution shape is an exponential of a negative square. That is precisely why the curve e^(−x²) appears so often in nature.

There is also another perspective worth mentioning: the normal distribution is the distribution of maximum entropy for a given mean and variance. But that’s a story for another time.