Probability Distribution: List of Statistical Distributions
The alpha distribution has been used for tool wear problems and may be used in modeling lifetimes under an accelerated test condition, even though the mean does not exist.
Johnson et al. [2] defines the probability density function (PDF) as:
Where Φ is the standard normal distribution.
There is a similarity between this PDF and that of the inverse normal density function. This is due fact that the PDF is just the density function of X = 1/Y when Y has a normally distributed random variable truncated to the left of zero with α = ξ , σ and β = 1/ σ [1].
NIST [1] defines the PDF slightly differently, in terms of the standard normal distribution CDF Φ and PDF φ:
Where α is the shape parameter.
Johnson et. al [1] describes the cumulative distribution function (CDF) of the alpha distribution as
The relationship between the CDF and the standard normal distribution is shown through basic integration [3].
The bathtub-shaped hazard rate function corresponding to Johnson’s PDF is [3]
Properties of the Alpha Distribution
Mean: Does not exist.
Mode:
- As α increases, the mode moves to the left.
- As β increases, the mode moves to the right.
- The beta alpha distribution [3] generalizes the alpha distribution.
References
[1] NIST. ALPPDF. Online: https://www.itl.nist.gov/div898/software/dataplot/refman2/auxillar/alppdf.htm
[2] Johnson, Kotz, and Balakrishnan, (1994), Continuous Univariate Distributions, Volumes I and II, 2nd. Ed., John Wiley and Sons.
[3] Corderiro et al. The Beta Alpha Distribution. Online: http://www.est.ufmg.br/portal/arquivos/rts/Beta_Alpha_Distribution_RT_UFMG.pdf