The second example is perhaps the most important pdf in probability and
statistics: the *Gaussian*, or *normal*, distribution.

This is a two-parameter ( and ) distribution, and it can be shown that is the mean of the distribution and is the variance. Figure 7 illustrates the Gaussian pdf.

Figure 7 Gaussian (Normal) Probability Distribution Function View figure

Let us calculate the probability that a sample from the Gaussian distribution will fall within a single standard deviation of the mean :

Similarly, the probability that the sample is within two standard deviations (within ``'') of the mean is

Hence 68% of the samples will, on average, fall within one , and over 95% of the samples will fall within two of the mean .

The Gaussian distribution will be encountered frequently in this course, not only because it is a fundamental pdf for many physical and mathematical applications, but also because it plays a central role in the estimation of errors with Monte Carlo simulation.