Notes on the sum and maximum of independent exponentially distributed random variables with different scale parameters

摘要

We consider the distribution of the sum and the maximum of a collection ofindependent exponentially distributed random variables. The focus is laid onthe explicit form of the density functions (pdf) of non-i.i.d. sequences. Thoseare recovered in a simple and direct way based on conditioning. A connectionbetween the pdf and a representation of the convolution characteristic functionas a linear combination of the single characteristic functions is drawn. It isdemonstrated how the results on the pdf of order statistics and the convolutionmerge.