Alpha distribution

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Graph of the alpha distribution PDF for various alpha values [1].

The alpha distribution is a continuous probability distribution often used to model non-negative data such as lifetimes, failure rates, and other similar datasets. Non-negative data refers to data values that are either positive (greater than zero) or equal to zero, meaning they do not include any negative values. This type of data is commonly found in situations where negative values are not possible or don’t make sense, such as measuring time, counting objects, or quantifying amounts like distance, weight, or size.

The 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.

Alpha distribution properties

The alpha distribution possesses several characteristics that render it valuable for modeling non-negative data. Firstly, as a heavy-tailed distribution, it features a long tail extending to infinity, making it suitable for modeling data with low probability of occurring. Secondly, its scale-free nature ensures that the shape remains consistent when the data is multiplied by a constant, making it ideal for modeling data measured in various units. In this context, “scale-free” refers to the property of a distribution that maintains its shape even when the data is multiplied by a constant, making it suitable for modeling data measured in different units without altering the underlying characteristics or relationships within the data.

This distribution is governed by two parameters: a shape parameter alpha and a scale parameter beta. The alpha parameter dictates the shape of the probability distribution, while the beta parameter determines its spread or size of the distribution on the horizontal axis.

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 (cut off) 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 cumulative distribution function (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]. Integration in calculus is a mathematical process that calculates the accumulated value or area under a curve by summing infinitesimally small elements.

The bathtub-shaped hazard rate function corresponding to Johnson’s PDF is [3]:

A hazard rate function is a statistical measure that represents the instantaneous probability of an event, such as failure or death, occurring at a specific time, given that the event has not occurred before that time.

Mean: does not exist.

The alpha distribution’s mean does not exist due to its heavy-tailed nature, with a long tail that indicates a small yet non-zero probability of observing values significantly larger than the mean. The alpha distribution has an especially heavy tail, making the probability of observing extremely large values much higher than in a normal distribution.

The mean of a probability distribution represents its expected value, calculated by taking the weighted average of all potential values, with the weight being the probability of observing that value. In a heavy-tailed distribution, the probability of observing very large values is non-zero, but their weights are minimal. Consequently, the distribution’s mean is dominated by the most probable values, those near the mean. With the alpha distribution, the probability of observing exceptionally large values is so minimal that their weights are virtually zero, resulting in an effectively zero mean, despite the small yet non-zero probability of observing very large values.

The absence of a mean in the alpha distribution can pose challenges for certain applications, such as estimating the expected value of costs or revenue streams, where the mean is often used. In the case of the alpha distribution, the mean is not a dependable estimate of the expected value, and alternative methods must be used.


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.

Alpha distribution uses

The alpha distribution finds uses in a diverse set of applications, such as:

  1. Modeling lifetimes: It is used to model the lifetimes of various products, components, and systems, including light bulbs, electronic components, and medical devices.
  2. Modeling failure rates: This distribution is used to represent the failure rates of systems and components, such as aircraft engines, power plants, and telecommunications systems.
  3. Modeling other non-negative data: The alpha distribution is also applied to model different non-negative data types, including insurance policy claims, defects in manufactured products, and expenditure on specific products or services.


[1] NIST. ALPPDF. Online:

[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:

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