Some properties of convolutions of Pascal and Erlang random variables

J. Mi, W. Shi and Y.Y. Zhou

Statistics & Probability Letters, 2008, vol. 78, issue 15, 2378-2387

Abstract: This article studies the convolutions of Pascal random variables, i.e., negative binomial distributions with integer parameters. We show that the probability distributions of such convolutions can be expressed as a generalized mixture (i.e., mixture with negative and positive coefficients) of a finite number of Pascal distribution functions. Based on this result, further study on the limiting behavior of the failure rate function of the convolution is presented. A sufficient condition is given to establish the likelihood ratio order between two convolutions of Pascal random variables. Similar results are obtained for the convolutions of Erlang random variables.

Date: 2008
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