The failure rate for the convolution of two distributions, one of which has bounded support

Abstract

We study the behavior of the failure rate associated with the distribution of a random variable of the form X = Y + U , where Y, U are independent and U has bounded support. First, we obtain monotonicity results and bounds for the failure rate of X in the case where U has a uniform distribution and, in particular we show that, asymptotically, the failure rates of X and Y tend to the same limit. Some of the results are generalized for the case where the distribution of U is not uniform, but has bounded support. Further, we show that if the failure rate of a non negative variable X is constant in some interval ( L , ∞ ) , then X can be written as the sum of two independent random variables, one of which is exponential and the other (which is not necessarily uniform) has support [ 0 , L ] .