## 2 The Pliant Probability Distribution Family  
In 2020 based on omega function [PERSON] and [PERSON] [21] proposed new four-parameter probability distribution function called the plant probability distribution function (see also ([22] Chapter 3)). **Definition 4**. : _The plant probability distribution function \(F_{p}(x;\alpha,\beta,\gamma,m)\) (pliant CDF) is defined by_  
\[F_{p}(x;\alpha,\beta,\gamma,m)=\left\{ \begin{array}{ll}0&\mbox{if }x\leq \frac{m}{2}(\gamma-1),\\ (1-\gamma\omega_{m}^{(-\alpha,\beta)}(x))^{\gamma}&\mbox{if }x\in(\frac{m}{2}( \gamma-1),m),\\ 1&\mbox{if }x\geq m,\end{array} \right. \tag{2}\]  
_where \(\omega_{m}^{(-\alpha,\beta)}\) is defined by (1) and \(\alpha,m\in\mathbb{R}\), \(\alpha>0\), \(m>0\), \(\beta\in\textbf{{B}}_{\gamma}\), \(\gamma\in\{-1,1\}\)._  
According to its properties, this new probability distribution can be applied in many fields of science and in a wide range of modeling problems. The plant probability distribution is a generalization of the epsilon probability distribution (see [23]). In 2020 [PERSON] noted that the omega probability distribution (see [20]) can be derived from the plant probability distribution function after reparametrization or by utilizing its asymptotic properties (see ([24] Lemma 2)). In 2020 [PERSON] [25] considered the asymptotic behavior of Hausdorff distance between the shifted Heaviside function and the so-called epsilon probability distribution. Once more he proved a precise bound for the values of Hausdorff distance. He noted that once may formulate the corresponding approximation problem for the omega probability distribution. This is the main purpose of Section 3 of this work. This section is dedicated to the behavior of the CDF function of the plant probability distribution and more precisely \"saturation to the horizontal asymptote \(a=1\) in the Hausdorff sense\". Let \(\alpha,\beta,m>0\), \(\gamma\in\{-1,1\}\) and \(t\in(\frac{m}{2}(\gamma-1),m)\). For the function \(F_{p}(x;\alpha,\beta,\gamma,m)\) given in (2) we have  
\[F_{p}(t_{0};\alpha,\beta,\gamma,m)=\frac{1}{2}\quad\text{with}\quad t_{0}=m \left(\frac{1-z}{1+z}\right)^{\beta-1},\quad z=\left(\frac{1}{\gamma}-\frac{2 ^{-1/\gamma}}{\gamma}\right)^{2(am^{\beta})^{-1}}. \tag{3}\]  
Then the Hausdorff distance \(d\) between \(F_{p}(x;\alpha,\beta,\gamma,m)\) and the Heaviside function \(h_{t_{0}}(t)\) satisfies the following nonlinear equation  
\[F_{p}(t_{0}+d;\alpha,\beta,\gamma,m)=1-d.\]  
In the next theorem, we prove upper and lower estimates for the Hausdorff approximation \(d\). **Theorem 1**. : _Let_  
\[A_{p}=1+\frac{1}{4}\alpha\beta\gamma m^{\beta-1}(1-2^{-1/\gamma})2^{-\frac{ \gamma-1}{\gamma}}\left(\frac{1-z^{2}}{z}\right)\left(\frac{1-z}{1+z}\right)^ {\beta-1} \tag{4}\]  
_and \(2.1A_{p}>e^{1.05}\) with \(z\) defined by (3). Then for the Hausdorff distance \(d\) between shifted Heaviside function \(h_{t_{0}}(t)\) and the Pliant CDF function \(F_{p}(t;\alpha,\beta,\gamma,m)\) defined by (2) the following inequalities hold true:_  
\[d_{l}=\frac{1}{2.1A_{p}}<d<\frac{\ln\left(2.1A_{p}\right)}{2.1A_{p}}=d_{r}.\]  
Proof. : Let us consider the function  
\[H(d)=F_{p}(t_{0}+d;\alpha,\beta,\gamma,m)-1+d,\]  
where the pliant CDF function \(F_{p}(t;\alpha,\beta,\gamma,m)\) is defined by (2). It is easy to show that \(H^{\prime}(d)>0\), so the function \(H(d)\) is increasing. We examine the following approximation of \(H(d)\) as we use the function  
\[G(d)=-\frac{1}{2}+A_{p}d,\]  
where \(A_{p}\) is given by (4). Indeed from Taylor expansion, we get \(G(d)-H(d)=\mathcal{O}(d^{2})\). This means that \(G(d)\) approximates \(H(d)\) with \(d\to 0\) as \(\mathcal{O}(d^{2})\) (see Figure 1). More over \(G^{\prime}(d)>0\) and function \(G(d)\) is also increasing. Let the following condition \(2.1A_{p}>e^{1.05}\) holds. Then it is easy to show that  
\[G(d_{l})=-\frac{1}{2}+A_{p}\frac{1}{2.1A_{p}}<0\quad\text{and}\quad G(d_{r})=- \frac{1}{2}+A_{p}\frac{\ln\left(2.1A_{p}\right)}{2.1A_{p}}>-\frac{1}{2}+\frac{ 1.05}{2.1}=0.\]  
This completes the proof.