Inequalities for sums of independent geometrical random variables
The concepts of majorization and Schur convexity are examined in relation to the probability distribution function of the sum of independent geometric and negative binomial variables. It is shown in particular, that the function \(P(X_1+ X_2+\cdots+ X_n\leq t)\), where \(X_1, X_2,\dots, X_n\) are independent geometric random variables with parameters \(p_1,p_2,\dots, p_n\) is Schur-concave with respect to \((p_1,p_2,\dots, p_n)\) for every real \(t\). The result is extended to the case of the negative binomial distribution, where each of the random variables \(X_i\), \(i= 1,2,\dots, n\), represents the number of trials until the \(r\)th success in a sequence of Bernoulli trials. The case where the random variables \(X_i\), \(i=1,2,\dots, n\), are distributed independently according to the negative binomial distribution with unequal parameters \(r_i\), \(i= 1,2,\dots,n\), is also considered.
geometric and negative binomial variables
scientific article; zbMATH DE number 1053606