2.4.2 Gaussian (Normal) Distribution:

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Next: 2.4.3 Cauchy Distribution Up: 2.4 Examples of Continuous Previous: 2.4.1 Exponential Distribution

2.4.2 Gaussian (Normal) Distribution:

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.