Please can someone explain the mathematical definition given for the residual holding time in the CR. {R_s>w,X_s=i}={X_u=i,s<u<s+w}. I understand the concept but the definition given seems a bit confusing. Also would i be correct in saying that the current holding time plus the residual holding time has an exponential distribution in a time-homogeneous Markov jump process since it is simply the total holding time?
thanks goku..you must be a super saiyan when it comes to CT4. that does make sense although i would be happier if the definition was {R_s=w,X_s=i}={X_u=i,s<u<s+w}, but sure its not worth getting upset about.
Gee thanks Floydeon, however I very much doubt I'll ever reach that level! Anyway, as regards your preference "{R_s=w,X_s=i}={X_u=i,s<u<s+w}", from my understanding of statistics and probability, that event occurs with probability zero. Well, it's because R_s has a continuous state space and therefore has a corresponding probability density function associated to it. Kinda like how f(x) is a density for a Normal distribution. So if indeed we wanted to find P[X = a] in this case, we would have to integrate the pdf from a to a, which is just zero. However a discrete distribution is associated with a probability mass function and is characterized by a 'jumping' non-continuous distribution. An example of that is N(t) where N(t) is Poi(lambda*t). Hence it is permissible to write and speak of P[N(t) = a]. So to conclude, to have X_u = i, s<u<s+w, we require the event {R_s > w,X_s=i} to occur. Hope that answers you query. Ka-me-ha-me-ha!
