Beta normal distribution

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While many common distributions like the normal, exponential, and Poisson distributions have been extensively studied, some lesser-known distributions offer unique insights and advantages in specific contexts. In this article, we will explore the beta normal distribution from different angles such as its shape properties, estimation methods, and practical applications.

What is the beta normal distribution?

The beta normal distribution, first introduced by Eugene et al. in 2002 [1], is a continuous probability distribution that results from combining two distributions: the beta distribution and the normal distribution.

  • The beta distribution is a popular distribution for modeling proportions, rates, and proportions since it can take values between 0 and 1 and has different shapes depending on the values of two parameters.
  • The normal distribution, on the other hand, is well-known for its bell-shaped probability density function (PDF) that describes many real-world phenomena, especially when the data are symmetric and unimodal.

By combining these two distributions, the beta-normal distribution inherits some of their properties and offers greater flexibility in modeling complex data distributions, including skewed, bimodal, and heavy-tailed ones.

Beta normal distribution properties

A random variable X has a beta normal distribution BN(α, β, μ, σ) with PDF [1]

beta normal distribution PDF


  • φ ((x – μ) / σ) = the normal distribution PDF
  • Φ ((x – μ) / σ) = the normal distribution CDF.
  • α, β = shape parameters,
  • μ = location parameter.
  • σ = scale parameter (stretches or shrinks the distribution).

One way to visualize the properties of the beta-normal distribution is to plot its density function for different combinations of beta and normal parameters.

The beta normal distribution can be either bimodal or unimoda
The beta normal distribution can be either bimodal or unimodal — depending on the parameters [2]

Some interesting properties of the beta normal distribution:

  • Symmetry about μ when α = β.
  • When α > 0.214 and β > 0.214, the distribution is always unimodal [3].
  • When α < 0.214 and β < 0.214, the distribution is always bimodal.

For unimodal distributions, the following properties hold

  • Right skewed when α > β.
  • Left skewed when α < β.
  • Platykurtic and symmetric when α < 1, β < 1 and α = β .
  • Leptokurtic when α > 1, β > 1.

Once we understand the shape properties of the beta-normal distribution, we can also estimate its parameters using statistical methods like the maximum likelihood method. The maximum likelihood estimation (MLE) is a powerful and widely used method for finding the values of distribution parameters that make the observed data most probable. In the case of the beta-normal distribution, the MLE involves finding the optimal values for the beta parameters α, β , and the normal parameters μ, σ, that maximize the likelihood function based on the data sample. The likelihood function is the product of beta and normal density functions evaluated at each data point. The MLE values can be obtained numerically using software packages like R or Python, and their accuracy can be assessed using goodness-of-fit measures like the Kolmogorov-Smirnov test or the Akaike information criterion.

Beta normal distribution applications

The beta normal distribution has many practical applications in different fields where the data have complex and diverse distributions. For instance, in finance, the beta-normal distribution has been used to model stock returns [4], where the beta component captures the market risk, and the normal component captures the asset-specific risk. In biology, the beta-normal distribution has been used to model gene expression data [5], where the beta component captures the fraction of genes that are differentially expressed, and the normal component captures the magnitude of the expression differences. Other applications include psychometric testing [6], crop yield [7] and risk assessment [8].

In conclusion, the beta-normal distribution is a powerful and flexible distribution that combines the beta and normal distributions, providing greater modeling capabilities for complex data distributions. By understanding its shape properties, we can visualize different types of beta-normal distributions and interpret their parameters. By using the maximum likelihood method, we can estimate the parameters from data samples and evaluate their accuracy. By applying the beta-normal distribution to different empirical data sets, we can demonstrate its usefulness and effectiveness in various fields where traditional distributions may fall short. As college students learning statistics or related fields, exploring different distributions like the beta-normal distribution can enrich your toolkit and enhance your problem-solving skills.


[1] Eugene, Nicholas & Lee, Carl & Famoye, Felix. (2002). Beta-normal distribution and its application. Communications in Statistics-theory and Methods – COMMUN STATIST-THEOR METHOD. 31. 497-512. 10.1081/STA-120003130.

[2] L. C. Rêgo, R. J. Cintra, G. M. Cordeiro. On Some Properties of the Beta Normal Distribution. Communications in Statistics – Theory and Methods, Volume 41, Issue 20, 2012 arXiv:2206.00762v1

[3] Eugene, N. A Generalized Normal Distribution: Properties, Estimation
and Applications. Unpublished Doctoral Dissertation: Central
Michigan University, Mount Pleasant, Michigan, 2001

[4] Sarabia Alegría, José María. Prieto Mendoza, Faustino. Jordá Gil, Vanesa. Some models of beta-generated distributions with applications in finance.

[5] Kang S, Song J. Robust gene selection methods using weighting schemes for microarray data analysis. BMC Bioinformatics. 2017 Sep 2;18(1):389. doi: 10.1186/s12859-017-1810-x. PMID: 28865426; PMCID: PMC5581932.

[6] Ranger J and Kuhn J-T (2015) Modeling Information Accumulation in Psychological Tests Using Item Response Times. Journal of Educational and Behavioral Statistics 40 274–306. [Google Scholar]

[7] Hennessy DA (2009) Crop Yield Skewness Under Law of the Minimum Technology. American Journal of Agricultural Economics 91 197–208. [Google Scholar]

[8] Jones MC (2004) Families of distributions arising from distributions of order statistics. Test, 13 1–43. [Google Scholar]

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