<p>The current research paper investigates the logistic <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(q-\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>q</mi> <mo>-</mo> </mrow> </math></EquationSource> </InlineEquation>distribution <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(q-\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>q</mi> <mo>-</mo> </mrow> </math></EquationSource> </InlineEquation>L(0,&#xa0;1) from a probabilistic perspective. Its unique characteristics lie in its probability density function (pdf), cumulative distribution function (cdf), mean, and variance. We introduce the <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(q-\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>q</mi> <mo>-</mo> </mrow> </math></EquationSource> </InlineEquation>Fourier transform and a new <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(q-\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>q</mi> <mo>-</mo> </mrow> </math></EquationSource> </InlineEquation>differential equation within the framework of quantum calculus and provide its solution in order to facilitate the characterization of the logistic <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(q-\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>q</mi> <mo>-</mo> </mrow> </math></EquationSource> </InlineEquation>distribution. A characterization of the relationship between the logistic <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(q-\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>q</mi> <mo>-</mo> </mrow> </math></EquationSource> </InlineEquation>distribution and exponential <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(q-\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>q</mi> <mo>-</mo> </mrow> </math></EquationSource> </InlineEquation>distributions is determined via the cumulative distribution function (cdf).</p>