If lambda is to be that small, then the force of mortality becomes negative, which to me doesnot makes any sense. Although, mathematically we are getting that answer, but i dont think that the solution makes any sense then.
Hi Kunjesh, Can you please give the actual question? I agree a negative force of mortality does not make sense by itself. It sounds like a question where we've used an adjusted force based on interest rates or something, so that the maths still stands up but I'd need to see the question, JOhn
Here is the screenshot of that question: https://drive.google.com/file/d/0BxIsAmUN3-LvRFNLdF80MkRjdGc/view?usp=sharing
Hi Kunjesh, Yes, you're right, with that formula, we must have lambda > 1/60, or mu could go negative. We can see this make sense again when we answer part (ii) The question isn't very good because it doesn't tell us whether it is an annual force of mortality or a daily one. (The English in it is pretty bad too, it's missing the word "the" in 2 places!) Let's assume the forces are daily... We have: 90q0 = 0.95 90p0 = 0.05 30p0 *60p30 = 0.05 exp{-0.03*30}*exp{-integral from 0 to 60 of (-0.5 + lamda*x dx)} = 0.05 exp(-0.9)*exp{-(-0.5x + 0.5 lambda*x^2) evaluated between 0 and 60} = 0.05 exp(-0.9)*exp{30 - 1800 lambda} = 0.05 So 60p30 = exp{30 - 1800 lambda} would be negative if lambda < 1/60. I think the question is fine but it would have been helpful to state that lambda > 1/60 or make this a part of the question, Good luck! John
N OK. But Sir, I believe that lambda should be greater than 1/2. Because, it has been given in the question that x is the number of days after day 30 and not the number of days that the mosquito lives. so the minimum value that x could possibly take is 1 making lambda to be necessarily greater than 0.5! And, so the integration should be 30 to 60 rather than 0 to 60.
Hi Kunjesh, Mmm... I see what you mean. I think I'm starting to agree with you that the question doesn't make sense to begin with! Even with lambda = 0.5, we can still have x=0.5 days and a negative force of mortality! The question only works if x is the number of days in total, NOT number of days since age 30. In this case, you're right that we should be integrating from 30 to 90 and NOT from 0 to 60. Let's hope you don't get another question like this in the IAI exams! John
