Derivation of the Normal distribution
The normal distribution, or normal density, is well known as the bell shaped curve. It has wide use in probability and statistics. In standard textbooks its definition is given as a two-parameter family of curves:
the parameter $\mu$ determines the location of its peak and the parameter $\sigma$ the fatness of the distribution. The standard normal distribution has $\mu=0$ and $\sigma=1$.
The first occurence of the normal distribution appeared in an article  in 1733 by Abraham de Moivre as an approximation to the Binomial distribution. We show de Moivres derivation in modern terminology. Suppose that the discrete random variable $X_n$ has a binomial distribution $X \sim B(n,p)$ with parameters $n=1,2,3,..$ and $0\less p\less1$ then $X$ has following probability mass function (pmf):
A discrete random variable has a binomial distribution if:
- An experiment, or trial, is performed in exactly the same way n times
- Each of the n trials has only two possible outcomes. One of the outcomes is called a "success," while the other is called a "failure." Such a trial is called a Bernoulli trial.
- The n trials are independent.
- The probability of success, denoted p, is the same for each trial. The probability of failure is q = 1 - p.
- The random variable X = the number of successes in the n trials.
When we draw the histogram of a binomial distribution we see that the distribution increase from the left to a maximum value, and then decreases. For increasing values of $n$ we see that the maximum value is lower, but shifts to the right. The value of $x$ which corresponds with the peak is the expected value.
Friederich Gauss derived the standard normal distribution as the probability distribution for random errors.
-  S. Stahl, The Evolution of the Normal Distribution, Mathematics Magazine 1996.
-  De Moivre, Normal Approximation to the Binomial Distribution, 1733, in A History of Probability and Statistics and Their Applications before 1750.
-  Vassilios C. Hombas, Deriving the normal density as a solution of a differential equation,University of Athens.
-  Dan Teague, The Normal Approximation: derivation from basic principles.,The North Carolina School of Science and Mathematics
-  Dan Teague, Approximatio ad Summam Terminorum Binomii (a + b)n in Seriem expansi ,12 November 1733 for private circulation, an English translation was incorporated into the 1738 (pp. 235-242) and the 1756 (pp. 243-250) editions of the Doctrine of Chances,an edited version of this article here
-  Pollard, Normal approximation to the BionomialLecture notes
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