f(x) should give the chance of something happening, thats the reason for f(inf)=<1. i used f(x)=1-exp(-a*x) until now, which is ok but f'(0) is not +inf. i would prefer a function with f(0)=+inf. i want the function to be (monotonically) increasing at a decreasing rate (f' decreasing monotonically from +inf to 0) does a function like this exist? many thanks in advance!