Some Intrinsic Properties of the Gamma Distribution

Abstract

Let {Yn} be a sequence of nonnegative random variables (rvs), and Sn = n j=1 Yj, n ≥ 1. It is first shown that independence of S k−1 and Y k , for all 2 ≤ k ≤ n, does not imply the independence of Y1, Y2, . . . , Yn. When Yj's are identically distributed exponential Exp(α) variables, we show that the independence of S k−1 andIt is shown by a counterexample that the converse is not true. We show that if X is a non-negative integer valued rv, then there exists, under certain conditions, a rvwhere {N (t)} is a standard (homogeneous) Poisson process, and obtain the Laplace-Stieltjes transform of Y . This leads to a new characterization for the gamma distribution. It is also shown that a G(α, k) distribution may arise as the distribution of S k , where the components are not necessarily exponential. Several typical examples are discussed.

On the Sums of Compound Negative Binomial and Gamma Random Variables

“…They mentioned that the distribution of such a sum is not expressible in a closed form and so discussed approximations and studied tail probabilities. Using the connection between the NB and gamma distributions (see Engel and Zijlstra (1980) or Vellaisamy and Sreehari (2008)), we obtain, in Section 3, the exact distribution of weighted sums of independent gamma random variables with arbitrary parameters.…”

“…Also, there exist (see Vellaisamy and Sreehari (2008)) independent standard Poisson processes {N j (t)} 1≤j ≤n and {N(t)} such that , β), where t and β are as defined in the theorem and L n is the discrete random variable with P(…”

“…Note that Engel and Zijlstra (1980)), where {N(t)} is a standard (parameter unity) Poisson process. Also, there exist (see Vellaisamy and Sreehari (2008)…”

On the Sums of Compound Negative Binomial and Gamma Random Variables

“…They mentioned that the distribution of such a sum is not expressible in a closed form and so discussed approximations and studied tail probabilities. Using the connection between the NB and gamma distributions (see Engel and Zijlstra (1980) or Vellaisamy and Sreehari (2008)), we obtain, in Section 3, the exact distribution of weighted sums of independent gamma random variables with arbitrary parameters.…”