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.
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 .
Johnson et. al  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 .
Properties of the Alpha Distribution
- As α increases, the mode moves to the left.
- As β increases, the mode moves to the right.
- The beta alpha distribution  generalizes the alpha distribution.
 NIST. ALPPDF. Online: https://www.itl.nist.gov/div898/software/dataplot/refman2/auxillar/alppdf.htm
 Johnson, Kotz, and Balakrishnan, (1994), Continuous Univariate Distributions, Volumes I and II, 2nd. Ed., John Wiley and Sons.
 Corderiro et al. The Beta Alpha Distribution. Online: http://www.est.ufmg.br/portal/arquivos/rts/Beta_Alpha_Distribution_RT_UFMG.pdf