# Inequalities for a distribution with monotone hazard rate

## Summary

Let*X* be a positive random variable with the survival function\(\bar F\) and the density*f*. Let*X* have the moments μ=E(*X*) and μ_{2}=E(*X*^{2}) and put ε=|1-μ_{2}/2μ^{2}|. Put\(q(x) = f(x)/\bar F(x)\) and\(q_1 (x) = \bar F(x)/\int_x^\infty {\bar F(u)du} \). It is proved that the following inequalities hold:\(|\bar F(x) - e^{ - x/\mu } | \leqq \varepsilon /(1 - \varepsilon e)\), for all*x*>0, if*q(x)* is monotone and that\(\int_0^\infty {|\bar F(x) - e^{ - x/\mu } |} dx \leqq 2\varepsilon \mu \), if*q*_{1}*(x)* is monotone. It is also shown that Brown's inequality\(|\bar F(x) - e^{ - x/\mu } | \leqq \varepsilon /(1 - \varepsilon )\) which holds whenever*q*_{1}*(x)* is increasing is not valid in general when*q*_{1} is decreasing.

## Key words

Characterization exponential distribution hazard rate mean residual life## Preview

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