28 Data Reduction and Error Analysis for the Physical Sciences There are several derivations of the Gaussian distribution from first principles, none of them as convincing as the fact that the distribution is reasonable, that it has a fairly simple analytic form, and that it is accepted by convention and experimentation to be the most likely distribution for most experiments. In addition, it has the satisfy- ing characteristic that the most probable estimate of the mean from a random sam- ple of observations x is the average of those observations x. Characteristics The Gaussian probability density is defined as * (2.23) This is a continuous function describing the probability of obtaining the value x in a random observation from a parent distribution with parameters and o, corre- sponding to the mean and standard deviation, respectively. Because the distribution is continuous, we must define an interval in which the value of the observation x will fall. The probability density function is properly defined such that the proba- bility dPG(x; , o) that the value of a random observation will fall within an inter- val dr PG 1 - 0V/, CP (-1(--)) V2 exp 2