HyperStudy

Statistical Distributions

Statistical Distributions

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Statistical Distributions

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Input variables can be characterized statistically using various statistical distributions. An input variable, when used in a statistical sense, is termed as a random variable. In ordinary usage, the term "random variable" indicates that the value this variable will take is unknown, but in a statistical sense, it is precisely known what values this variable will take and the probability associated with that value.

Input variables exhibit different properties depending on the parameter they represent. Some variables may be symmetric about the mean value, while others may be skewed towards either the left or right. Some variables may be bounded on either side or unbounded.

The first categorization of random variables is whether the variable is continuous or discrete. A random variable is considered continuous if it can assume any value in a given interval. A random variable is termed discrete if it can only assume a finite set of values within a given interval.

Input variables can be characterized using the following statistical distributions:

hmtoggle_plus1Exponential

Use Exponential distribution to describe the amount of time between occurrences, mean time between failures.                                          

 

exp_prop_density_plot

exp_prop_density

exp_cum_density

where lambda is the scale parameter.

 

 
hmtoggle_plus1Log-Normal

Use Log-Normal distribution in risk analyses.

hs_lognormal

sd4;

where m and s are location and scale.

 

 
hmtoggle_plus1Normal (CoV or Variance)

Use Normal (CoV or Variance) distribution to approximate many phenomenons in nature.

normaldist3

sd1;

where y is the mean and beta is the standard deviation.

In HyperStudy, a normal distribution can be defined using mean, y and variance, variance1 or using mean, y and coefficient of variance (CoV), beta/y.

Variance is the second statistical moment and measures the spread of a distribution. CoV measures the relative spread of a distribution. The higher the CoV, the higher the variability is.

 

 
hmtoggle_plus1Triangular

Use Triangular distribution when the only known information is the minimum, the most likely, and the maximum values.                                

triangular

triangulardist;

where lowera, lowerb and lowerc are the end points and the mode.

 

 
hmtoggle_plus1Uniform

Use Uniform distribution when all values between the minimum and maximum are equally likely such as a number from a random number generator.

uniformdist

 

sd2;

where lowera and lowerb are end points.

 

 
hmtoggle_plus1Uniform Discrete

Use Uniform Discrete distribution when you have discrete (numeric or string) variables that take values which are equally likely.

 

Example

Possible numeric values are 1, 2, 3, or 4; each are equally likely.

Or

Possible string variables are "orange", "green", "red", or "blue"; each are equally likely.

distribution_uniform_discrete

 
hmtoggle_plus1Weibull

Principal applications are situations involving wear, fatigue and failure, failure rates, life-time expectancies.

weibulldist ;

where alpha and b are shape and scale parameters which enable it to be adjusted to desired fatigue or reliability laws.