November 2002 # 2

[FSAwannabe]

For a triple decrement table you are given:
u(1)(t)=0.3
u(2)(t)=0.5
u(3)(t)=0.7
Calculate q(2).

My question is that there is no assumptions given about the distribution of the decrements. It seems that you go about solving the problem by assuming uniform distribution in the multiple decrement context. According to ARCH a more common assumption is uniform distribution in the associated single decrement table which would lead to a different formula. Right?

I am also not sure why we cant just figure this out with the following formula (which does not lead to the right answer):
q(j) = u(j)*P(r) {where p(r) = survival of all decrements}


Thanks for the help,

[Colymbosathon ecplecticos]

Are those forces of mortality? If so, you have all the information that you need.

[FSAwannabe]

yes those are forces of mortality.. can you please explain.
Thanks

[Colymbosathon ecplecticos]

UDD is an assumption that you might make if you knew the yearly mortality, but not how you got there.

Forces are continuous things --- you don't need to make any assumptions about what happens between them.

[FSAwannabe]

So why doesn't this give me the correct answer?
q(j) = u(j)*P(r) {where p(r) = survival of all decrements}

[Woody]

You need to find the integral from 0 to 1 of tpx(tau)*mux(2)dt, where tpx(tau) is exp^integral form 0 to 1 of -(0.3+0.5+0.7)t or exp^(-1.5t).

Doing this integral from 0 to 1 of exp^(-1.5t)*(0.5)dt gives you 0.26

[skookie]

There is also a shortcut you can use when dealing with "constant force of mortality."

mu^(tau) = .3+.5+.7 = 1.5
px^(tau) = e^-1.5 = .22313
qx^(tau) = 1-.22313 = .77687

notice that since the sources of decrement are all constant force, the probability of decrement due to cause 2 is always 1/3 the total probability of decrement. Since this ratio never changes throughout the year, the probability of decrement due to cause 2 throughout the year is 1/3 the total probability of decrement for the year...so:

qx^(2) = (1/3)*.77687 = .25896