Distribution Functions

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Beta Distribution

You provide the shape parameters r and t and the lower and the upper limit xmin and xmax of the random variable x.

The Beta distribution is very useful for random variables that are bounded at both sides. If linear operations are applied to random variables that are all subjected to a uniform distribution, then the results can usually be described by a Beta distribution. For example, if you are dealing with tolerances and assemblies where the components are assembled and the individual tolerances of the components follow a uniform distribution (a special case of the Beta distribution), the overall tolerances of the assembly are a function of adding or subtracting the geometrical extension of the individual components (a linear operation). Hence, the overall tolerances of the assembly can be described by a Beta distribution. Also, as previously mentioned, the Beta distribution can be useful for describing the scatter of individual geometrical extensions of components as well.

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Exponential Distribution

You provide the decay parameter λ and the shift (or lower limit) xmin of the random variable x.

The exponential distribution is useful in cases where there is a physical reason that the probability density function is strictly decreasing as the uncertainty variable value increases. The distribution is mostly used to describe time-related effects; for example, it describes the time between independent events occurring at a constant rate. It is therefore very popular in the area of systems reliability and lifetime-related systems reliability, and it can be used for the life distribution of non-redundant systems. Typically, it is used if the lifetime is not subjected to wear-out and the failure rate is constant with time. Wear-out is usually a dominant life-limiting factor for mechanical components that would preclude the use of the exponential distribution for mechanical parts. However, where preventive maintenance exchanges parts before wear-out can occur, then the exponential distribution is still useful to describe the distribution of the time until exchanging the part is necessary.

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Gaussian (Normal) Distribution

You provide values for the mean value μ and the standard deviation σ of the random variable x.

The Gaussian, or normal, distribution is a 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 speaking, every random variable follows a normal distribution if it is generated by a linear combination of a very large number of other random effects, regardless which distribution these random effects originally follow. The Gaussian 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.

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Lognormal Distribution

You provide values for the logarithmic mean value ξ and the logarithmic deviation δ. The parameters ξ and δ are the mean value and standard deviation of ln(x):

The lognormal distribution is another basic and commonly-used distribution, typically used to describe the scatter of the measurement data of physical phenomena, where the logarithm of the data would follow a normal distribution. The lognormal distribution is suitable for phenomena that arise from the multiplication of a large number of error effects. It is also used for random variables that are the result of multiplying two or more random effects (if the effects that get multiplied are also lognormally distributed). It is often used for lifetime distributions such as the scatter of the strain amplitude of a cyclic loading that a material can endure until low-cycle-fatigue occurs.

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Uniform Distribution

You provide the lower and the upper limit xmin and xmax of the random variable x.

The uniform distribution is a fundamental distribution for cases where the only information available is a lower and an upper limit. It is also useful to describe geometric tolerances. It can also be used in cases where any value of the random variable is as likely as any other within a certain interval. In this sense, it can be used for cases where "lack of engineering knowledge" plays a role.

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Triangular Distribution

You provide the minimum value xmin, the most likely value limit xmlv and the maximum value xmax.

The triangular distribution is most helpful to model a random variable when actual data is not available. It is very often used to capture expert opinions, as in cases where the only data you have are the well-founded opinions of experts. However, regardless of the physical nature of the random variable you want to model, you can always ask experts questions like "Out of 1000 components, what are the lowest and highest load values for this random variable?" and other similar questions. You should also include an estimate for the random variable value derived from a computer program, as described above. For more details, see Choosing a Distribution for a Random Variable.

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Truncated Gaussian Distribution

You provide the mean value μ and the standard deviation σ of the non-truncated Gaussian distribution and the truncation limits xmin and xmax.

The truncated Gaussian distribution typically appears where the physical phenomenon follows a Gaussian distribution, but the extreme ends are cut off or are eliminated from the sample population by quality control measures. As such, it is useful to describe the material properties or geometric tolerances.

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Weibull Distribution

You provide the Weibull characteristic value xchr , the Weibull exponent m and the minimum value xmin. Special cases: For xmin = 0 the distribution coincides with a two-parameter Weibull distribution. The Rayleigh distribution is a special case of the Weibull distribution with α = xchr - xmin and m = 2.

In engineering, the Weibull distribution is most often used for strength or strength-related lifetime parameters, and is the standard distribution for material strength and lifetime parameters for very brittle materials (for these very brittle material the "weakest-link theory" is applicable). For more details, see Choosing a Distribution for a Random Variable.