bibkey: sen1999geometricconvolution authors: A. Sen; N. Balakrishnan year: 1999 title: “Convolution of geometrics and a reliability problem” doi: 10.1016/S0167-7152(98)00284-3 claim: “Theorem 1 gives the distribution of a sum of independent non-identical geometric variables. For positive-integer summands with success probabilities 1−λ_i and distinct 0<λ_i<1, its survival function is P(T>n)=sum_i c_i λ_i^n, where c_i=product_{l≠i}(1−λ_l)/(λ_i−λ_l); hence P(T>n)=1 for 0≤n<q, where q is the number of summands.” strata_touched: [] license: citation-only triage: anchor

Geometric convolution and an initial survival plateau

For independent geometric variables supported on the positive integers, write

$$\mathrm{Pr}(G_i=k)=(1-{\lambda }_i){\lambda }_i^{k-1},T=\sum _{i=1}^q{G}_i,0<{\lambda }_i<1,$$

with pairwise distinct parameters. Theorem 1 yields the geometric-convolution distribution. In this convention its survival form is

$$\mathrm{Pr}(T>n)=\sum _{i=1}^q{c}_i{\lambda }_i^n,c_i=\prod _{\ell \ne i}\frac{1-{\lambda }_{\ell }}{{\lambda }_i-{\lambda }_{\ell }}(n\ge 0).$$

This expression is a survival probability. The corresponding probability mass is

$$\mathrm{Pr}(T=n)=\sum _{i=1}^q{c}_i(1-{\lambda }_i){\lambda }_i^{n-1}(n\ge 1).$$

Since every summand is at least one, the survival probability equals one for $0\le n<q$ and lies in $[0,1]$ for every $n\ge 0$. The tail-sum identity gives

$$\sum _{n\ge 0}\mathrm{Pr}(T>n)=\mathbb{E}T=\sum _{i=1}^q\frac{1}{1-{\lambda }_i}.$$

Verified locator

• A. Sen and N. Balakrishnan, Convolution of geometrics and a reliability problem, Statistics & Probability Letters 43(4) (1999), 421–426.
• DOI: https://doi.org/10.1016/S0167-7152(98)00284-3
• Result locator: Theorem 1; the formulas above use positive-integer geometric waiting times and express the resulting survival function.