DEMONSTRATIO MATHEMATICA Vol. XXXVI No 1 2003 Helena Jasiulewicz, Wojciech Kordecki* CONVOLUTIONS OF ERLANG AND OF PASCAL DISTRIBUTIONS WITH APPLICATIONS TO RELIABILITY Abstract. The main aim of the paper is to give a generalization of the results given by Sen and Balakrishnan in [7]. The results obtained are used to calculate the reliability of some class of systems. 1. Introduction The distribution of the sum of n independent exponentially distributed random variables with different parameters fn (i = 1,2, ..., n) is given in [2], [3], [4] and [7]. In this paper, we give the distribution of this sum without the assumption that all the parameters m are different. We use the fact that the sum of k independent identically distributed exponential random variables with parameter ¡ JL has an Erlang distribution with k degrees of freedom and parameter //, i.e. Erl ( k , fi). Thus, grouping the components of the sum which have the same parameter //, the problem reduces to one of finding the distribution of the sum of independent random variables having Erlang distributions. This problem will be considered in Section 2. All of the above problems are the special cases of the general case of a sum of independent random variables with gamma distributions. A for- mula for such a sum was provided by Mathai [5] in 1982. This formula is significantly complicated even in the case when random variables are expo- nentially distributed. Sen and Balakrishnan [7] describe it as "substantially messy" then derive it again in the particular case, when all random vari- ables are exponentially distributed and intensities are all distinct from each other. For completeness, the formulae provided by Mathai will be presented in Section 3. A similar problem for a sum of random variables with geometric distri- bution, where all the parameteres are different is considered in [7]. In this paper, we give the distribution of the sum of n independent random variables »Research partially supported by KBN grant 2 P03A 053 15.