Stochastic inequalities for weighted sum of two random variables independently and identically distributed as exponential

Let X and Y be two random variables which are independently and identically distributed (i.i.d.) as exponential. Given two nonnegative numbers a and b, it is of interest to establish bounds on the tail probability of aX+bY, i.e. P(aX+bY>t). The present work attempts to provide first some basic inequalities for P(aX+bY>t) and then shows that this probability increases in (a,b) defined on the set D={(a,b):a2+b2=1,a>b} for some t. This result is further supported and enhanced by numerical computation.