In this thesis we present some related problems about the renewal processes. More precisely, let $gamma_{t}$ be the residual life at time $t$ of the renewal process $A={A(t),t geq 0}$, $F$ be the common distribution function of the inter-arrival times. Under suitable conditions, we prove that if $Var(gamma_{t})=E^2(gamma_{t})-E(gamma_{t}),forall t=0,1 ho,2 ho,3 ho,... $, then $F$ will be geometrically distributed under the assumption $F$ is discrete. We also discuss the tails of random sums for the renewal process. We prove that the $k$ power of random sum is always new worse than used ($NWU$).