Dakota Reference Manual  Version 6.4
Large-Scale Engineering Optimization and Uncertainty Analysis
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normal_uncertain


Aleatory uncertain variable - normal (Gaussian)

Topics

This keyword is related to the topics:

Specification

Alias: none

Argument(s): INTEGER

Default: no normal uncertain variables

Required/Optional Description of Group Dakota Keyword Dakota Keyword Description
Required means First parameter of the distribution
Required std_deviations Second parameter of the distribution
Optional lower_bounds Specify minimum values
Optional upper_bounds Specify maximium values
Optional initial_point

Initial values

Optional descriptors

Labels for the variables

Description

Within the normal uncertain optional group specification, the number of normal uncertain variables, the means, and standard deviations are required specifications, and the distribution lower and upper bounds and variable descriptors are optional specifications. The normal distribution is widely used to model uncertain variables such as population characteristics. It is also used to model the mean of a sample: as the sample size becomes very large, the Central Limit Theorem states that the distribution of the mean becomes approximately normal, regardless of the distribution of the original variables.

The density function for the normal distribution is:

\[f(x) = \frac{1}{\sqrt{2\pi}\sigma_N} e^{-\frac{1}{2}\left(\frac{x-\mu_N}{\sigma_N}\right)^2}\]

where $\mu_N$ and $\sigma_N$ are the mean and standard deviation of the normal distribution, respectively.

Note that if you specify bounds for a normal distribution, the sampling occurs from the underlying distribution with the given mean and standard deviation, but samples are not taken outside the bounds (see "bounded normal" distribution type in[89]). This can result in the mean and the standard deviation of the sample data being different from the mean and standard deviation of the underlying distribution. For example, if you are sampling from a normal distribution with a mean of 5 and a standard deviation of 3, but you specify bounds of 1 and 7, the resulting mean of the samples will be around 4.3 and the resulting standard deviation will be around 1.6. This is because you have bounded the original distribution significantly, and asymetrically, since 7 is closer to the original mean than 1.

Theory

When used with design of experiments and multidimensional parameter studies, distribution bounds are inferred. These bounds are [ $\mu - 3 \sigma$, $\mu + 3 \sigma$]

For vector and centered parameter studies, an inferred initial starting point is needed for the uncertain variables. These variables are initialized to their means for these studies.