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.
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.