Past exam of the Probability, statistics and options II course of the University of Oxford / 2011 / M / M4 / d

For $t\ge 0$, the geometric survival probability givesFor $t<0$, the distribution functions are zero, as is the limiting exponential distribution function. Since that limiting distribution function is continuous everywhere, these limits proveThe parameter is rate one, so the limiting mean is one, consistent with $\mathbb{E}(S_1/k)=1$.