in this proof they have written nPxx = 1 - nQxx can anybody explain how nPxx = 1 - nQxx nPxx should be equal to 1 - (npx*nqx + nqx*npx + nqx*nqx)
Suggest you check back at the summary on page 38 of Chapter 8. nqxx is the probability that at least one of the lives dies, not that they both die.
but the point arise that how come they replace nPxx with 1 - nQxx is there any way to prove that npxx = 1 - nqxx npxx means at least one of the life die for example if we have two policy holders named x and y of same age and same mortality so npxx will include the following cases xdies*y survives + y dies* x survives + x dies * y dies
sorry I might be being dumb - but your equation is right too. The one they use is simply right by definition - no need for a mathematical proof. npxx means they both survive. If they don't, one must have died, and this is what nqxx is defined to be. If you put a bar over the lives it means both have died..
so what i can see here is they have given a complete notation of nqxx = (npx*nqx + nqx*npx + nqx*nqx) since nqxx means at least one dies and (npx*nqx + nqx*npx + nqx*nqx) also means at least one dies which makes this now nqxx + npxx = 1 and that is why npxx = 1 - nqxx