Questions in Ch19 and Ch22

[iamminime]

Hi, I have several questions in the CMP:
1. Ch19, P.5, second bullet point:
"the probability of x or y dying in the interval from time t to time t+dt ...", I think is [tPx*mu(x+t) + tPy*mu(y+t)]*dt ?
why it says it's [mu(x+t) + mu(y+t)]*dt ?

2. Ch22, P.21, there is an equation, d/dt t(aq)s:x = sigma*t(ap)x
When sigma is not constant, on P.34, d/dt t(aq)s:x = t(ap)x * sigma(x+t)
Where do these 2 equations come from?

3. Ch22, P.33, force of mortality according to the ELT15 (Males) mortality table is mu50 = 0.00440
Why in the solution it takes q50 and derive mu50 by -ln(1-q50)=0.00465 instead of just taking mu50 in ELT15?

Please help. Thanks a lot!

 

[iamminime]

Also, in Ch.25, P.5, at the bottom for Year 4
2108.81 - 105.44 + 80.13 - 80 - 84.35 = 1919.15, why is there 60 more?

 

[Joe Hook]

Hi,

1. Ch19, P.5, second bullet point:

The joint life status fails on the first death so we need both lives to survive to time t and then one of the lives to die. This is allowed for in the course notes as two separate probabilities multiplied together (the two bullet points). Your expression is close to being right but you've worked out the probability of either x or y dying at time t without any conditioning of the other life needing to be alive. Hence it doesn't quite work out the probability that the first death is happening at time t.

2. Ch22, P.21, there is an equation, d/dt t(aq)s:x = sigma*t(ap)x
When sigma is not constant, on P.34, d/dt t(aq)s:x = t(ap)x * sigma(x+t)
Where do these 2 equations come from?

Mathematically i'm not sure (likely CS2). From general reasoning the probability that a life leaves the active population via decrement s at time t is the probability they remain active to time t (ie survive all decrements to time t) multiplied by the force of transition to state s at time t.

3. Ch22, P.33, force of mortality according to the ELT15 (Males) mortality table is mu50 = 0.00440
Why in the solution it takes q50 and derive mu50 by -ln(1-q50)=0.00465 instead of just taking mu50 in ELT15?

mu50 from the tables is the force of mortality at EXACT age 50 ie from age 50 to 50 + dt. So as soon as the life ages, even by a millisecond, mu will change. We are more interested in using the constant force of mortality across a year of age and hence we derive the equivalent constant mu by using p50.

4. Also, in Ch.25, P.5, at the bottom for Year 4
2108.81 - 105.44 + 80.13 - 80 - 84.35 = 1919.15, why is there 60 more?

This appears to be a typo in our notes. Thanks for flagging. I agree your 1919.15.

Thanks
Joe