For , and , this probability density function has cumulative distribution function . Differentiating gives the density, while the endpoint limits are zero and one. Its quantile function is for , enabling inverse transform sampling.
Expanding the denominator as a geometric series givesTermwise integration, justified by the monotone convergence theorem, and the gamma function integral yield raw moments . Thus the polylogarithm expresses its mean, variance and higher moments. Here is the complement of the series parameter ; parameter conventions must be checked when comparing formulas.
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The exponential-logarithmic distribution is a probability distribution that combines elements of both exponential and logarithmic distributions. It's not as commonly discussed or as widely known as other distributions like the normal, exponential, or uniform distributions; therefore, details can be somewhat fragmented. The exponential-logarithmic distribution may refer to a specific use case or modification of well-known distributions in fields such as survival analysis, queuing theory, or reliability engineering.