HjorthDistribution

HjorthDistribution[m,s,f]

represents the Hjorth distribution with location parameter m, scale parameter s, and shape parameter f.

Details

  • HjorthDistribution is a distribution that has been applied in reliability analysis for modeling different classes of failure behavior.
  • The survival function for value in a Hjorth distribution is given by for and otherwise.
  • HjorthDistribution[m,s,f] allows m and s to be any real positive numbers and f to be any non-negative real number.
  • HjorthDistribution[m,s,f] allows m, s and f to be any quantities such that m*s and m*f are dimensionless. »
  • HjorthDistribution can be used with such functions as Mean, CDF, and RandomVariate.

Examples

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Basic Examples  (3)

Probability density function for different parameter values:

Cumulative distribution function for different parameter values:

Mean:

Variance is available numerically:

Scope  (6)

Generate a sample of pseudorandom numbers from a Hjorth distribution:

Compare its histogram to the PDF:

Distribution parameters estimation:

Estimate the distribution parameters from sample data:

Compare the density histogram of the sample with the PDF of the estimated distribution:

Different moments with closed forms as functions of parameters:

Moment:

Closed form for symbolic order:

CentralMoment:

FactorialMoment:

Cumulant:

Hazard function:

Quantile function:

Consistent use of Quantity in parameters yields QuantityDistribution:

Find median time:

Applications  (5)

The lifetime of a device follows HjorthDistribution. Find the reliability of the device:

Find the mean lifetime:

Plot the possible shapes of the hazard rate:

The failure behavior of a component is described by HjorthDistribution with parameters , and . Find the probability that the component fails within its first year:

Plot the density function of the time of failure:

Find the time at which the component is most likely to fail:

Find the mean time to failure:

Find how long a component can survive with a safety of 90%:

Find the probability that the component survives for at least one more year after having survived two years:

Simulate the failure times for 30 independent components like this:

A piece of electronic equipment has initially high failure rate due to the randomness in the quality variations during its production. Its lifetime can be modeled by HjorthDistribution with parameters , and . Plot the hazard function:

Mean lifetime:

Probability of failure within the first year (warranty period):

To avoid early failures, the equipment is operated at stress level during a "burn-in" period. Find the length of the burn-in period after which the failure probability within the first year is reduced by half:

Find the expected lifetime of the equipment after having survived the burn-in period:

Find the time at which the equipment is most reliable:

Simulate the failure times for 50 independent pieces of equipment like this:

A rather simple mechanical system is composed of three independent components: two of type A and one of type B. The measured failure times, in days, are:

Find the estimated distribution for both components, assuming HjorthDistribution:

Compare the distribution of the time to failure of each component with the data:

The system works as long as one component of each type is working. Find the reliability of the system:

Find the mean time to failure:

Find the probability that the system fails after three days:

A simple mechanical system is composed of three independent components: two of type A and one of type B. The system works as long as one component of each type is working. The failure times of each component type follow the distributions:

Find the probability that both components A fail before B does:

Each time that a component A fails, it is immediately replaced. Find the average number of components of type A used by the system before B fails:

The expected lifetime of the system when adding more components of type A:

Plot the results:

Properties & Relations  (5)

Hjorth distribution is closed under scaling by a positive factor:

Relationships to other distributions:

Hjorth distribution simplifies to RayleighDistribution:

Hjorth distribution can be obtained as ParameterMixtureDistribution of a linear hazard rate distribution with initial failure rate following GammaDistribution:

The linear hazard rate distribution could be obtained as OrderDistribution of ExponentialDistribution and RayleighDistribution:

ExponentialDistribution is a limiting case of Hjorth distribution:

Possible Issues  (5)

HjorthDistribution is not defined when m or s is not a positive real number:

HjorthDistribution is not defined when f is a non-negative real number:

Substitution of invalid parameters into symbolic outputs gives results that are not meaningful:

The characteristic function of the Hjorth distribution does not have a closedform representation:

The function values can be obtained for numeric inputs:

The closed form of the moments of Hjorth distribution could lead to some numerical instabilities for some parameters:

Find numerical approximation for the moment using inexact parameters:

Alternatively, evaluate the closed expression at higher precision:

Neat Examples  (1)

PDF for different s values with CDF contours:

Introduced in 2017
 (11.1)