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<p>We establish a criterion for the unimodality of the probability distribution of a functional that is represented by the sum of a set of independent identically distributed random nonnegative variables <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10958_2025_7881_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">\({\tilde{x}}_k\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mover accent="true"> <mi>x</mi> <mo stretchy="false">~</mo> </mover> <mi>k</mi> </msub> </math></EquationSource> </InlineEquation> with a random number of terms distributed according to Poisson. The general distribution of terms <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10958_2025_7881_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">\({\tilde{x}}_k\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mover accent="true"> <mi>x</mi> <mo stretchy="false">~</mo> </mover> <mi>k</mi> </msub> </math></EquationSource> </InlineEquation> is concentrated on the interval [0,&#xa0;1] and is such that Pr<InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10958_2025_7881_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="101" /> </InlineMediaObject> <EquationSource Format="TEX">\(\{{\tilde{x}}_k = 0\} \ne 0.\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">{</mo> <msub> <mover accent="true"> <mi>x</mi> <mo stretchy="false">~</mo> </mover> <mi>k</mi> </msub> <mo>=</mo> <mn>0</mn> <mo stretchy="false">}</mo> <mo>≠</mo> <mn>0</mn> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> Its absolutely continuous part is asymptotically close to a uniform distribution. We introduce the concept of smoothing functions and establish an explicit form of the distribution of any fixed number of terms uniformly distributed on [0,&#xa0;1].</p>

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UNIMODALITY OF THE PROBABILITY DISTRIBUTION OF THE EXTENSIVE FUNCTIONAL OF SAMPLES OF A RANDOM SEQUENCE

  • 期刊论文

  • 发表日期：

    2025年7月1日

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  • Yu. P. Virchenko,
  • A. M. Tevolde

摘要

We establish a criterion for the unimodality of the probability distribution of a functional that is represented by the sum of a set of independent identically distributed random nonnegative variables \({\tilde{x}}_k\) x ~ k with a random number of terms distributed according to Poisson. The general distribution of terms \({\tilde{x}}_k\) x ~ k is concentrated on the interval [0, 1] and is such that Pr \(\{{\tilde{x}}_k = 0\} \ne 0.\) { x ~ k = 0 } ≠ 0 . Its absolutely continuous part is asymptotically close to a uniform distribution. We introduce the concept of smoothing functions and establish an explicit form of the distribution of any fixed number of terms uniformly distributed on [0, 1].