can someone help me on the proof of the probablities of death or survival in chapter 4 page 5? i didn't get the first line of equation - is P(alive at x+t+dt|alive at x+t) the same as dtPx+t ? Can we use the condition probablity formular here? Also at the last step, to get the solution of the differential equation - why (d/dt(tPx))/(tPx)=d/dt(ln(tPx))? Thanks!
Question on survival prob Hi Annie, is P(alive at x+t+dt|alive at x+t) the same as dtPx+t ? Yes Can we use the condition probablity formular here? Yes, it's a conditional prob Why (d/dt(tPx))/(tPx)=d/dt(ln(tPx))? I bet you might not have asked this question if it was why d/dt(ln(tPx))=(d/dt(tPx))/(tPx) ! In my head, my A-level maths teacher still tells me "If we differentiate logs we get reciprocal times differentiate the inner bit". So, if we see a formula that is the derivative of something over itself we can say "oh that's the same as if we differentiate the log of the thing" Good luck! John
Thanks, John But i still don't get the proof - if i use condition probablity formular , i get t+dtPx=tPx*dtPx+t (since P(alive at x+t+dt|alive at x+t) is the same as dtPx+t ), where does the second part about transition probablity come from?
Transition probabilities over small time periods Hi Annie, The probability of staying alive for a time period dt (ie dtpx) is equal to dt times px + o(dt). The o(dt) term is an error term to show that the equation is not exact. But the smaller dt becomes, the smaller o(dt) becomes and the approximation gets better. When we take limits later in the algebra, we are no longer making an approximation, Good luck! John
