## 3 Omega Probability Distribution  
Using the omega function [PERSON] et al. [20] defined the so-called Omega probability distribution. In the next definition, we give the corresponding CDF function.  
**Definition 5**: _The Omega probability cumulative distribution function \(F(x;\alpha,\beta,m)\) (omega CDF) is defined by_  
\[F(x;\alpha,\beta,m)=\left\{\begin{array}{ll}0&\mbox{if }x\leq 0,\\ 1-\omega_{m}^{(-\alpha,\beta)}(x)&\mbox{if }0<x<m,\\ 1&\mbox{if }x\geq m,\end{array}\right. \tag{5}\]  
_where \(\omega_{m}^{(-\alpha,\beta)}\) is defined by (1) and \(\alpha,\beta,m\in\mathbb{R}\), \(\beta,m>0\), \(x\in(0,m)\)._  
Figure 3: The Pliant probability distribution function \(F_{p}(t;\alpha,\beta,\gamma,m)\) for different values of parameters and corresponding Hausdorff distance \(d\).  
In this section, we investigate the omega probability distribution in Hausdorff sense as a continuation of the work of [PERSON] [25] and as a corollary of the pliant probability distribution.  
Let \(\alpha,\beta,m>0\) and \(t\in(0,m)\). For the function \(F(x;\alpha,\beta,m)\) given in (5) we have  
\[F(t_{0};\alpha,\beta,m)=\frac{1}{2}\quad\text{with}\quad t_{0}=m\left(\frac{2^{ 2(\alpha m^{\beta})^{-1}}-1}{2^{2(\alpha m^{\beta})^{-1}}+1}\right)^{\beta^{-1 }}.\]  
Then the Hausdorff distance \(d\) between \(F(t;\alpha,\beta,m)\) and the Heaviside function \(h_{t_{0}}(t)\) satisfies the following nonlinear equation  
\[F(t_{0}+d;\alpha,\beta,m)=1-d\]  
or  
\[\left(\frac{m^{\beta}+(t_{0}+d)^{\beta}}{m^{\beta}-(t_{0}+d)^{\beta}}\right)^ {\frac{\alpha m^{\beta}}{2}}=\frac{1}{d}.\]  
The next theorem is a corollary of Theorem 1 in the case of the omega distribution.  
**Theorem 2**.: _Let_  
\[A=1+\frac{1}{8}\alpha\beta m^{\beta-1}\left(\frac{4^{2(\alpha m^{\beta})^{-1}} -1}{4^{(\alpha m^{\beta})^{-1}}}\right)\,\left(\frac{4^{(\alpha m^{\beta})^{ -1}}+1}{4^{(\alpha m^{\beta})^{-1}}-1}\right)^{\beta^{-1}}\]  
_and \(2.1A>e^{1.05}\). Then for the Hausdorff distance \(d\) between shifted Heaviside function \(h_{t_{0}}(t)\) and the Omega CDF function \(F(t;\alpha,\beta,m)\) defined by (5) the following inequalities hold true:_  
\[d_{l}=\frac{1}{2.1A}<d<\frac{\ln{(2.1A)}}{2.1A}=d_{r}. \tag{6}\]  
In Table 3 we present several computational examples to show behavior of the Omega CDF function \(F(t;\alpha,\beta,m)\) with different values of parameters \(\alpha,\beta\) and \(m\). We use Theorem 2 for computation of values of upper and lower estimates \(d_{l}\) and \(d_{r}\). It can be seen that the proven bottom estimates for the value of Hausdorff distance \(d\) is reliable in approximation of shifted Heaviside function \(h_{t_{0}}(t)\) and the Omega CDF function \(F(t;\alpha,\beta,m)\). Graphical representation in Figure 4 shows that the important characteristic \"saturation\" is faster.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(\alpha\) & \(\beta\) & \(m\) & \(d_{l}\) & \(d\) & \(d_{r}\) & Figure 4 \\ \hline
0.87 & 0.98 & 4.51 & 0.330061 & 0.361465 & 0.365865 & (a) \\
0.86 & 0.97 & 6.56 & 0.333677 & 0.365581 & 0.366238 & (b) \\
0.33 & 1.88 & 3.85 & 0.328542 & 0.350978 & 0.365697 & (c) \\
0.54 & 3.11 & 7.84 & 0.238728 & 0.259147 & 0.341961 & (d) \\
3.01 & 0.39 & 2.13 & 0.067058 & 0.180506 & 0.181204 & (e) \\
2.35 & 0.53 & 8.76 & 0.166522 & 0.254183 & 0.298512 & (f) \\
2.64 & 3.21 & 4.81 & 0.177190 & 0.198820 & 0.306634 & (g) \\
3.33 & 2.89 & 7.46 & 0.174810 & 0.198154 & 0.304879 & (h) \\ \hline \hline \end{tabular}
\end{table}
Table 3: Some bounds for Hausdorff distance \(d\) by (6).