Hi, I can’t understand part of the formulae for a survival probability on this page of the notes. About midway down the page it goes from : “(t+dt)p(x) = (t)p(x) * (1 - mu(x+t)dt + o(dt))” to saying: “therefore : d/dt (t)p(x) = lim (dt > 0) (((t+dt)p(x)) - (t)p(x)) / dt) thanks in advance
Hi Harj By definition: d/dt tpx = lim(dt -> 0+) ((t+dt)px - tpx) / dt Rearranging the first equation you quoted gives: ((t+dt)px - tpx) = - tpx mu(x+t) dt + o(dt) tpx Substituting this into the derivative definition above, we get: d/dt tpx = lim(dt -> 0+) {- tpx mu(x+t) dt + o(dt) tpx} / dt We can write this as: d/dt tpx = lim(dt -> 0+) {- tpx mu(x+t) + o(dt) / dt} by noting that tpx o(dt) is just some other o(dt) function. The first part of the RHS doesn't depend on dt and so can be taken out of the limit, this gives: d/dt tpx = - tpx mu(x+t) + lim(dt -> 0+) {o(dt) / dt} Hope this helps! Andy
