Hello > > > > I have a question in regards to the CT5 notes. > > On page 15 of chapter 13, it states: > > > > (aq)_x^a = ... = integral_o^1 (al)_(x+t)...dt > > > > a==alpha > > > > can you please explain which idenity or rules are being used to get the > > second and third equality signs. > > > > i.e. for the 2nd equality somehow (ad)_x^a = integral_0^1 (al)_(x+t)(a > > mu)_(x+t)^a dt > > why is this the case? > > > > and similarly what relationship is used to get the next equality?
Hi Amaster - You're a few chapters ahead of me there, but l'll have a go based on CT4... Please ignore if this doesn't make any sense! For the first: ad_x^a is the number dying (or running away with a swedish model or whatever else decrement a is) between ages x and x+1. So that's the integral_0^1 of the intensity of deaths at age x+t To get the intensity of deaths, we need to multiply the active population (al)_(x+t) by the force of mortality (a mu)_(x+t)^a . For the second: t_(ap)_x =(al)_(x+t)/(al)_x as probability of remaining active for time t= number active at x+t/number active at x - i.e. more or less by the definition of (al)_(x+t) and we've assumed (a mu)^a_(x+t)=(mu)^a_(x+t) ("assumption of equal forces..."). Then they multiply and divide by t_p_x^a for the next step of the calc. Happy xmas! Louisa
