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82 • using ansys for finite eLement anaLysis
e − x 2 2 /
fx () =
2p
Figure 3.2 shows the standard normal pdf.
3.2.2.2 Cumulative Distribution Function
The formula for the cdf of the normal distribution does not exist in a simple
closed formula. It is computed numerically. Figure 3.3 shows the normal cdf.
3.2.2.3 Common Statistics
Mean The location parameter m
Median The location parameter m
Mode The location parameter m
Range Infinity in both directions
Standard Deviation The scale parameter s
/
Coefficient of Variation sm
Skewness 0
Kurtosis 3
3.2.2.4 Parameter estimation
The location and scale parameters of the normal distribution can be esti-
mated with the sample mean and sample standard deviation, respectively.
3.2.2.5 Comments
The Gaussian or normal distribution is a very fundamental and commonly
used distribution for statistical matters. It is typically used to describe the
scatter of the measurement data of many physical phenomena. Strictly speak-
ing, every random variable follows a normal distribution if it is generated by
a linear combination of a very large number of other random effects, regard-
less of which distribution these random effects originally follow. The Gauss-
ian distribution is also valid if the random variable is a linear combination of
two or more other effects if those effects also follow a Gaussian distribution.
For both theoretical and practical reasons, the normal distribution is
probably the most important distribution in statistics. For example:
