1Blurb:: 2Aleatory uncertain variable - normal (Gaussian) 3 4Description:: 5The number of normal uncertain variables, their means, and standard 6deviations are required specifications, while the distribution lower and 7upper bounds and variable descriptors are optional specifications. The 8normal distribution is widely used to model uncertain variables such 9as population characteristics. It is also used to model the mean of a 10sample: as the sample size becomes very large, the Central Limit 11Theorem states that the distribution of the mean becomes approximately 12normal, regardless of the distribution of the original variables. 13 14The density function for the normal distribution is: 15\f[ f(x) = \frac{1}{\sqrt{2\pi}\sigma} 16 \exp \left(-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2 \right) \f] 17where 18\f$\mu\f$ and \f$\sigma\f$ 19are the mean and standard deviation of the normal distribution, respectively. 20 21Note 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 \cite Wyss1998). 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. 22 23Topics:: continuous_variables, aleatory_uncertain_variables 24Examples:: 25Theory:: 26When used with some methods such as design of experiments and 27multidimensional parameter studies, distribution bounds are inferred 28to be [\f$\mu - 3 \sigma\f$, \f$\mu + 3 \sigma\f$] 29 30For some methods, including vector and centered parameter studies, an 31initial point is needed for the uncertain variables. When not given 32explicitly, these variables are initialized to their means, correcting 33to bounds if needed. 34 35Faq:: 36See_Also:: 37