The beta distribution family gained popularity a few years ago, and various studies have been conducted on beta compounding with other distributions such as Beta-Frechet [1], Beta-Gumbel [2], Beta-Pareto [3], Beta-Gamma [4], Beta-Gompertz [5], Beta-Normal[6], Beta-Power [7], Beta-Weibull [8], Beta-Exponentiated Weibull [9], Beta-Modified Weibull [10], Beta-Extended Weibull[11], the Beta-Generalized Logistic [12], Beta-Log-Logistic [13], Beta-Exponential [14], Beta-Generalized Exponential [15], Beta-Moyal [16], Beta-Generalized Half-Normal [17], Beta-Dagum [18], Beta-Laplace [19], Beta-Burr XII [20], Beta-Generalized Pareto [21], Beta-Cauchy [22], Beta-Half-Cauchy [23], Beta-Exponentiated Pareto [24], Beta-Log-Logistic [13], Beta-Type I Generalized Half Logistic [25], Beta-Bessel Distributions [26], Beta-Rayleigh Distribution [27], Beta-Exponentiated Lindley [28], Beta-Lindley [29], Beta-Lindley-Poisson Distribution [30], and Beta-Lindley Geometric Distribution [31].

This study presents a five-parameter distribution called the Beta-Exponential-Gaussian distribution, which extends the Ex-Gaussian distribution by adding two additional parameters to control its skewness and kurtosis.

In cognitive psychology research, the Ex-Gaussian distribution is used to examine the semantic stroop effect [32], reading eye movements [33], and response time distributions [34]. Additionally, it is utilized to mimic fixation lengths [35], which are frequently employed in eye-tracking research as a gauge of cognitive processes.

This investigation unfolds to examine the new Beta-Ex-Gaussian distribution. In Section 2, a parent sub-model, the basic Ex-Gaussian distribution is introduced. The transformation technique for the proposed distribution is explained in Section 3. Methodically, Section 4 examines the Beta-Ex-Gaussian distribution. Advancing to Section 5, visual representations of the probability density function (PDF) and cumulative distribution function (CDF) of the Beta-Ex-Gaussian distribution are presented. Section 6 investigates statistical distinctions by deriving expressions for moments. The discourse within Section 7 explores the sphere of order statistics. Survival and hazard functions are scrutinized in Section 8. In Section 9, attention is directed toward estimation methodologies, with an emphasis on maximum likelihood. Section 10 offers simulation results from a comprehensive acceptance-rejection process. Section 11 expounds on four practical applications. Finally, Section 12 provides a conclusive conclusion, thereby concluding the article.

The exponential-Gaussian (Ex-Gaussian), also called the exponentially modified Gaussian (EMG) distribution, is a probability distribution that convolutions exponential and normal random variables and is used in signal processing, finance, and neuroscience for skewness and heavy tails data. It has a closed-form probability density function and a cumulative distribution function, making it useful for statistical analysis [36–39]. For ∀μ ∈ ℝ, σ, λ > 0, and ∀x ∈ ℝ, its PDF, CDF, Hazard, and Survival functions are given in the Equations (1–4), respectively.

where erfc is the complementary error function defined by erfc(x)=2π∫x∞e-t2dt.

Many transformation methods were applied to generate new probability distributions. This study uses the beta-generated (B-G) approach to generate continuous probability distributions. This technique modifies a base distribution using a generator function, resulting in a variety of shapes and characteristics. The beta distribution is used as the generator, adding more parameters to fit different shapes and determining skewness by the sum of all forms [40].

Let f(x; Θ) and F(x; Θ) be the probability density function and the cumulative distribution function (cdf) of a random variable X, respectively, where Θ is a p × 1 parameter vector, and then the cumulative distribution function is generated by applying [6, 29]

where α > 0 and β > 0 are two additional parameters whose role is to introduce skewness and vary the tail weight. B(α,β)=∫01tα-1(1-t)β-1dt=Γ(α)Γ(β)Γ(α+β) is the beta function, B(y;α,β)=∫0ytα-1(1-t)β-1dt is the incomplete beta function with B(α, β) = B₁(α, β), and Iy(α,β)=B(y;α,β)B(α,β) is the incomplete beta function ratio. The corresponding probability density function of Equation (5) is then given as follows:

This family of distributions can be considered as generalization of the distribution of order statistics [40] for the random variable X with CDF F(x). When α and β are integers, Equation (6) is the α^th order statistic of the random sample of size (α+β−1).

Let F(x; Θ) be the baseline CDF of an Ex-Gaussian, a continuous random variable, with Θ = (μ, σ, λ) as parameter vectors. We now introduce the five-parameter Beta-Ex-Gaussian (BExG) distribution by taking G(x) from Equation (5), the CDF of Equation (6). Substituting F(x; Θ) in Equation (5) by the CDF Equation (2) yields the CDF of the BExG as follows:

where α > 0 and β > 0 are new parameters that controls the skewness and kurtosis of the distribution, i.e., the distribution's shape, and ∀μ ∈ ℝ, σ, λ > 0, and ∀x ∈ ℝ are the location, scale, and rate parameters, respectively. A random variable X with the CDF Equation (7) is said to have a BExG distribution and will be denoted by X~BExG(Φ), where Φ = (μ, σ, λ, α, β).

For any values of α and β, we can write Equation (5) in terms of the well-known hypergeometric function (see [1, 5, 7]) given as follows:

Theorem 1. The new random variable X for x ∈ ℝ, λ > 0, μ ∈ ℝ, σ > 0, α > 0, and β > 0 having cumulative distribution function (CDF) and probability density function (PDF) expressed as follows:

where  2F1(a,b,c,z)=∑k=0∞((a)k(b)k)((c)kk!)zk and (a)_k = a(a+1)⋯(a+k−1).

Proof.

This function, g(x) in (2a), is always non-negative, which is trivial to prove, and

To prove (2b) integrates is 1 over the real number, we used Equation (6) for g(x)

Use integration by substitution: u = F(x) and differentiate u with respect to x, yielding:

The integral ∫0∞uα-1(1-u)β-1du is a special case of the beta function, defined as:

Table 1 shows that some closed-form expansions for Equation (5) distribution based on real, non-integer, or integer parameters.

Now, by changing variables, let t=u1-u we get u=t1+t. To change limit of integration from 0 to ∞, consider t → 0, u → 0;t → 1, u → ∞.

Therefore, we have:

Therefore, we have shown that ∫0∞g(x)dx=1

Corollary 1.1. Asymptotic properties:

Proof. To prove (i), the property for the confluent hypergeometric function ₂F₁ from the book [41] is:

where Γ(·) is the Gamma function and the properties of the beta function, respectively.

Then,

We have demonstrated conclusively that:

To prove (ii)

Therefore, limx→∞g(x)=0.

In this section, the BExG probability density function (PDF) and cumulative distribution function (CDF) plots in the Figures 1–5 are presented, showcasing various selected parameter values.

A bimodal PDF (Figure 5) shows two different peaks or modes, indicating the possibility of two unique processes or sub-populations with various traits. The relative heights of each peak indicate the frequencies and separations between them, while each peak itself represents a mode or cluster within the data.

Some of the shape properties of the uni-modal Beta-Ex-Gaussian distribution include:

In general, the new distribution provides more flexible and versatile shapes that are different from those of normal and exponential-normal distributions.

In probability and statistics, expectations of powers are moments of random variables, where the first is the expectation and the second is the variance, or the second central moment [43].

For β ∈ ℝ\ℤ, we have the power series

If β ∈ ℤ, the index j in the sum stops at β−1 [7].

For α ∈ ℝ\ℤ, F(x)^α+j−1 can be expanded:

Theorem 2. When α and β are integers, the n^th moment of the Beta-Ex-Gaussian random variable BExG(α, β, μ, σ, λ) is given in the Equation (13) as follows:

where

Proof. The book [44] explains that for a random variable X in Lⁿ, the n^th moment, the mean, and the central moment respectively are:

Using Equations (11, 12), and Karr [44] we can prove it as follows:

where

The variance, skewness, and kurtosis measures can be calculated and related using the study mentioned in Jafari et al. [5] and Bury [43] in the Equations (14–16) respectively.

The skewness and kurtosis measures are controlled mainly by the parameters α and β, and Figure 6 illustrates their variation using μ = 0, σ = 1, and λ = 1.

The skewness and kurtosis measures are controlled mainly by the parameters α and β, and Figures 6–10 illustrates their variation using various parametric values for μ, σ, and λ. More figures are illustrated in Appendix A. The skewness and kurtosis demonstrate strictly increasing, strictly decreasing, and U-shaped behaviors, which are interesting in some applications of the model.

Moreover, Tables 2, 3 illustrate how the mean and median values of a distribution offer insights into its central tendency. Skewness gauges the asymmetry of the distribution, with positive values suggesting longer or fatter right tails and negative values indicating the opposite. Kurtosis, on the other hand, measures the tailedness of the distribution, with higher values indicating heavier tails or more outliers. Additionally, the impact of α on the distribution's skewness and kurtosis can be observed; a higher α value may result in a more asymmetric distribution.

The density of the i^th order statistic X_i:n, g_i:n(x) say, in a random sample of size n from the BExG distribution is obtained from the well-known formula [7] is given in the Equation (17) as follows.

Or let X₁, X₂, ..., X_n be a random sample of size n from BExG(μ, σ², λ, α, β). Then the pdf and cdf of the i^th order statistic, say X_i:n, are given by Jafari et al. [5]

respectively, where Gi+m(x)=(∑r=0∞brF(x)r)i+m. Here, we use an equation by Gradshteyn and Ryzhik [45] and Jafari et al. [5], for a power series raised to a positive integer n

where the coefficients c_{n, r} (for r = 1, 2, ...) are easily determined from the recurrence Equation (20)

where cn,0=b0n. The coefficient c_{n, r} can be calculated from c_{n, 0}, ..., c_{n, r−1} and hence from the quantities b₀, ..., b_r.

The Equations (18, 19) can be written as follows:

Therefore, the s^th moment of X_i:n is as follows

which cannot be an explicit expression.

In this section, we refer to Klein and Moeschberger [46] to underscore the fundamental significance of the likelihood of an individual surviving beyond time x as a key metric in characterizing time-to-event events. The survival function, denoted by the random variable X, is described in the Equation (21) as follows (see Klein and Moeschberger [46] for details).

Furthermore, the hazard function, also referred to as the conditional failure rate, is a critical parameter in survival analysis with broad applications in fields such as economics, epidemiology, demography, and stochastic processes. For a continuous random variable X, the hazard function is defined as follows:

Theorem 3. The survival function and hazard function for the new random variable are as follows:

Proof. Equations (21, 22) can be used to easily obtain the proof for Equations (23, 24), respectively.

Corollary 3.1. Asymptotic behaviors:

limx→∞S(x)=0

Proof.