New results on stochastic comparisons of two-component series and parallel systems

Let X1,X2, and X3 be independent random variables with absolutely continuous distributions having the common support [0,∞). We show that if X1≤hr[mrl,lr]X3 and X2≤hr[mrl,lr]X3, then max{X1,X2}≤hr[mrl,lr]max{X1,X3}. We also show that if X2≤rh[lr]X1 and X2≤rh[lr]X3, then min{X1,X2}≤rh[lr]min{X1,X3}. These results generalize and extend some of the results given in Shaked and Shanthikumar (2007, Example 1.C.36, p. 56), Joo and Mi (2010), and Da et al. (2010).