The beta distribution has traditionally been employed in the PERT methodology and generally used for modeling bounded continuous random variables based on expert's judgment. The impossibility of estimating four parameters from the three values provided by the expert when the beta distribution is...

The beta distribution has traditionally been employed in the PERT methodology and generally used for modeling bounded continuous random variables based on expert’s judgment. The impossibility of estimating four parameters from the three values provided by the expert when the beta distribution...

We consider the mixed AR(1) time series model <Equation ID="Equa"> <EquationSource Format="TEX">$$X_t=\left\{\begin{array}{ll}\alpha X_{t-1}+ \xi_t \quad {\rm w.p.} \qquad \frac{\alpha^p}{\alpha^p-\beta ^p},\\ \beta X_{t-1} + \xi_{t} \quad {\rm w.p.} \quad -\frac{\beta^p}{\alpha^p-\beta ^p} \end{array}\right.$$</EquationSource> </Equation>for −1  β <Superscript> p </Superscript> ≤ 0 ≤ α <Superscript>...</superscript></superscript></equationsource></equation>

Two-sided confidence intervals for a probability <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$p$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi>p</mi> </mrow> </math> </EquationSource> </InlineEquation> under a prescribed confidence level <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$\gamma $$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi mathvariant="italic">γ</mi> </mrow> </math> </EquationSource> </InlineEquation> are an elementary tool of statistical data analysis. A confidence interval has two basic quality characteristics: i) exactness, i. e., whether the actual coverage...</equationsource></equationsource></inlineequation></equationsource></equationsource></inlineequation>