How Abraham de Moivre Discovered the Normal Distribution

追加: 2026-05-05

[00:00:00]In statistics, we have been using a particular distribution to model the probability of random variables. Normal distribution. Where does it come from?

[00:00:09]It was discovered by a French mathematician in a London coffee house in 1733 trying to help gamblers calculate binomial probabilities.

[00:00:20]This mathematician is Abraham Demo. He fled France due to religious persecution and ended up in London. To make a living, he calculated odds for gamblers.

[00:00:33]One day, a gambler brought him a problem. If you flip a fair coin 3,600 times, what is the probability of getting exactly 1,800 heads or between 1780 and 1820 heads?

[00:00:50]The exact probability for 1,800 heads of 3600 flips comes from the binomial distribution. It gives us this fraction.

[00:01:00]The problem here is the factorial symbol. 3,600 factorial means 3,600 * 3599 * 3,598 all the way down to 1. Calculating a number this massive by hand is impossible. Demo needed a more efficient method.

[00:01:21]He started by graphing the probabilities for smaller numbers of flips like two or 10 or 40.

[00:01:30]He noticed that as the number of flips increases, the discrete bar chart can be approximated by a continuous symmetric curve. He realized that if he could find the equation for this continuous curve, he could use calculus to find the area underneath it rather than adding up thousands of discrete fractions.

[00:01:50]To find the curve's shape, Demo analyzed how the probability drops off as you move x steps away from the peak by setting up a ratio of the probabilities and expanding the binomial factorials.

[00:02:04]He canceled out the common terms to leave a product of fractions. He then divided every term by n /2 to make the values closer to one.

[00:02:14]Working with massive products is difficult. So he took the natural logarithm to turn the fractions into addition and subtraction. Using the tailaylor series approximation that the natural log of 1 + z is roughly z for small values. The terms simplify.

[00:02:30]Factoring out -2 /n gives a simple addition of integers. The arithmetic sum formulas simplify perfectly into x^2 leaving exactly -2x^2n.

[00:02:43]Since the variance n * p * q for a fair coin is n / 4, the exponent matches exactly with the variance. Finally, taking the exponential of both sides reveals that the shape of the binomial probabilities smoothly approaches this exponential curve.

[00:03:02]With the curve's exponential shape determined, Demo had the formula y = c * ex^2 over 2n pq. He still needed to find the exact peak height c. The peak happens exactly at the middle of the coin flips.

[00:03:20]For even n the probability is n factorial / n / 2 factorial 2 * 2 ^ of n. For n is odd in that case there are two equal peaks at the center. But as n gets larger the difference between these two denominators becomes negligible. The limit remains exactly the same as the even case. To solve this he used an approximation for massive factorials.

[00:03:48]While Demora had discovered the core formula, he was stuck on finding the exact constant. He reached out to his friend James Sterling who identified the missing piece as the square root of 2 pi. Demo graciously credited Sterling for completing the discovery. Let's substitute Sterling's formula into the peak probability. The numerator n factorial expands and the denominator n /2 factorial is squared. When we square the denominator terms, the square root disappears and the exponent doubles.

[00:04:24]Notice how the 2 to the^ of n cancels out with the two in the denominator's fraction. This leaves us with n / e to the n on both the top and bottom which cancel each other out completely. We are left with the square<unk> of 2 /<unk> * n. Since n * p * q for a fair coin is n / 4, we finally get our constant c.

[00:04:49]This gave demo the complete function. He discovered the normal distribution, the single most important curve in statistics.

[00:04:59]To solve the gamblers's problem, we calculate the probability of getting exactly 1,800 heads. However, the area of a single line under a continuous curvaceous is exactly zero. Since we are approximating discrete coin flips, we must apply a continuity correction. The physical bar for exactly 1,800 actually spans from 1,799.5 to 1,800.5.

[00:05:27]By integrating over this one unit width, we find the probability is approximately 0.0133.

[00:05:34]With this correction, Demo elegantly solved the Gambler's problem using continuous calculus. Comparing the two, the normal approximation is incredibly accurate, providing a near perfect shortcut for complex binomial calculations.

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