The probability that I am smiling at any given point during the day is given by a function f(t) [note this is not a pdf but simply the probability of the event at a given time; thus 0<=f(t)<=1 but int_0^24 f(s) ds does not necessarily equal 1].
What is the probability P that I will smile during a specific time interval[t1,t2] during the day?
I have noted that:
int_t1^t2 f(s) ds has units time and is not bounded above by 1;
breaking the time interval into smaller timesteps and treating the probability in each of these as independent (and then multipliying) gives arbitary results since it is dependent on the choice of time step;
taking the mean or maximum etc. of f(t) in the interval would mean that watching me for one minute or twenty four hours could have the same probability of success;
obviously if f(t)=1 for any t in[t1,t2] then P=1 while if f(t)=0 for all t in[t1,t2] then P=0.