I have read countless pages on google and cannot find a satisfactory answer. I have also read http://castatistics.wikispaces.com/file/view/normal+der..pdf, but I doubt that was the original motivation for the Gaussian function. I am currently an undergraduate and my textbook just tells me the function f(x) = a e -(x - b)^2/c is used as the probability density function for a normal curve. But my textbook gives me no clues as to where this function actually came from. What was the original motivation for the development of such a function? Can someone please offer a proof that I can actually undertsand with clearly labeled steps? I have a undertsanding of basic calculus and I am a beginner when it comes to statistics. Please no complicated proofs.
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2$\begingroup$ In response to a very similar question, I offered stats.stackexchange.com/a/3904, which you might find useful. Strictly speaking, it does not answer your question about "motivation": that came several generations after de Moivre, with Gauss' development of the method of least squares. $\endgroup$– whuber ♦Commented Oct 22, 2012 at 14:36
2 Answers
The normal distribution is the distribution that is expected when measurements are made up from a large number of 'noise' components that are all distributed in the same way as each other.
The principle is sometimes illustrated with an example using dice. Throw one die a large number of times and plot the distribution of values. Assuming the die is fair, you will end up with a (discrete) uniform distribution from 1 to 6. Now do that again but use two dice. You get a stepwise triangular distribution from 2 to 12. Add a third die and the distribution is a little bit bell-shaped and the steps are small because there are now 17 different possible values. With four dice the distribution looks very much like a normal distribution, and with an infinite number of dice it is a normal distribution. Somewhere between four and an infinite number of die (I often say 12) are needed for a distribution that is, for practical purposes, indistinguishable from the normal distribution given by the normal formula.
Many biological and physical measurements have lots of sources of inaccuracy and noise and so the distributions of those measurements will be approximately normal, as long as the distributions of those components is similar. If one noise component is much larger than the others then the normal distribution will not result. Imagine if one die out of a set of dozen had faces marked from 100 to 600 rather than 1 to 6. That die would dominate the other eleven and so the distribution of the sum of their top faces would be an obvious mixture of (discrete) uniform 100 to 600 and a nearly continuous nearly normal 11 to 66. The distributions of the component variations have to be similar, even if they don't need to be normal (they don't have to be even nearly normal if there are a lot of them).
(It is worth noting that many sources of variability have a logarithmic distribution and so many measurements in biology and physics are more nearly log-normal than normal.)
The original derivation came from de Moivre who used it as an approximation to the binomial. It was later derived independently several times in other contexts.
http://en.wikipedia.org/wiki/Abraham_de_Moivre#Probability
http://en.wikipedia.org/wiki/De_Moivre%E2%80%93Laplace_theorem
