Bayrakdar, Kuş and Ng [Journal of Quality Technology, 2026, 1–18, doi: 10.1080/00224065.2025.2612362] introduced the shape-extended continuous Bernoulli distribution and derived series representations for its rth raw moment, its moment generating function and, under the restriction <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\gamma _1 \!=\! \gamma _2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>γ</mi> <mn>1</mn> </msub> <mspace width="-0.166667em" /> <mo>=</mo> <mspace width="-0.166667em" /> <msub> <mi>γ</mi> <mn>2</mn> </msub> </mrow> </math></EquationSource> </InlineEquation>, its stress-strength reliability <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\theta \!=\! \Pr (X < Y)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>θ</mi> <mspace width="-0.166667em" /> <mo>=</mo> <mspace width="-0.166667em" /> <mo>Pr</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo><</mo> <mi>Y</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. Each of these representations involves one or more infinite sums that must be truncated for practical use, raising questions of convergence, accuracy and computational efficiency. In this note, we resolve all three issues simultaneously by expressing each quantity in terms of well-known special functions. Specifically, we show that the rth raw moment and the moment generating function admit exact closed forms involving the confluent hypergeometric function <InlineEquation ID="IEq3"> <EquationSource Format="TEX">${}_1F_1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mmultiscripts> <mrow /> <mn>1</mn> <mrow /> </mmultiscripts> <msub> <mi>F</mi> <mn>1</mn> </msub> </mrow> </math></EquationSource> </InlineEquation> and the Srivastava-Daoust generalized Kampé de Fériet function <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathscr {S}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">S</mi> </math></EquationSource> </InlineEquation>, respectively, both of which are implemented to arbitrary precision in all major computer-algebra systems. We also derive a closed form expression for the stress-strength reliability <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\theta = \Pr (X < Y)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>θ</mi> <mo>=</mo> <mo>Pr</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo><</mo> <mi>Y</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> that places no restriction on the shape parameters <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\gamma _1$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>γ</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\gamma _2$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>γ</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation>, thereby extending the result of Bayrakdar et al. (<CitationRef CitationID="CR1">2026</CitationRef>).

Closed form Expressions for Moments, Moment Generating Function and Stress–strength Reliability of the Shape–extended Continuous Bernoulli Distribution

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发表日期：

2026年7月3日

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Saralees Nadarajah,
Tibor K. Pogány

摘要

Bayrakdar, Kuş and Ng [Journal of Quality Technology, 2026, 1–18, doi: 10.1080/00224065.2025.2612362] introduced the shape-extended continuous Bernoulli distribution and derived series representations for its rth raw moment, its moment generating function and, under the restriction $\gamma _1 \!=\! \gamma _2$ $γ_{1} = γ_{2}$ , its stress-strength reliability $\theta \!=\! \Pr (X < Y)$ $θ = Pr (X < Y)$ . Each of these representations involves one or more infinite sums that must be truncated for practical use, raising questions of convergence, accuracy and computational efficiency. In this note, we resolve all three issues simultaneously by expressing each quantity in terms of well-known special functions. Specifically, we show that the rth raw moment and the moment generating function admit exact closed forms involving the confluent hypergeometric function ${}_1F_1$ $_{1} F_{1}$ and the Srivastava-Daoust generalized Kampé de Fériet function $\mathscr {S}$ $S$ , respectively, both of which are implemented to arbitrary precision in all major computer-algebra systems. We also derive a closed form expression for the stress-strength reliability $\theta = \Pr (X < Y)$ $θ = Pr (X < Y)$ that places no restriction on the shape parameters $\gamma _1$ $γ_{1}$ and $\gamma _2$ $γ_{2}$ , thereby extending the result of Bayrakdar et al. (2026).