Survival Model

[Avviey]

Hi,

Can someone have a look at Question 7.8 on page 18 from Chapter 7, the Survival Model?

As it is given from the formula that

ex = 1Px + 2Px +…..+ w-sPx = 1P x + ∑ k=2 kP x = 1Px + ∑ j=1 1P x *
j Px+1 = ……..

I am wondering if it should be

ex = 1Px + 2Px +…..+ w-sPx = 1P x + ∑ k=2 kP x = 1Px + ∑ j=1 1P x *
j q x+1 = ……..

Because if you look at working of ex, you will see that its ∑ j Px * q x+j instead of ∑ j Px * Px+j.

Thanks very much for the help.

Regards,

Avv.

 

[Giles]

Avv

When you say the working of ex, I assume you mean the proof that ex = ∑k=1 kPx at the top of page 18.

From the acted bit below the core reading, it does give that kPx = ∑ j Px * q x+j, but this is summing from j = k to j = [w-x], not from j = 1, so I don't think that bit's relevant (at least not the way I answered the question)

ex = 1Px + 2Px +…..+ w-sPx = 1P x + ∑ k=2 kP x

If you look at the summation, it's 2Px + 3Px + ... + [w-x]Px

You can factor 1Px out of this by noting that 2Px = 1Px * 1P[x+1], 3Px = 1Px * 2P[x+1] etc, up to [w-x]Px = 1Px * [w-x-1]P[x+1]. This is pretty intuitive - the probability that a 22 year old lives 10 years is the same as the probability that they live one year times the probability that a 23 year old lives 9 years.

You can then rewrite this as a summation of 1Px * jP[x+1] for j = 1 to [w - x -1], then factor out the 1Px as in the solution to get the answer.

Hope that makes sense & helps - if the notation doesn't make sense then message me and I'll email you a word doc with proper formulae.

Giles

 

[Avviey]

It makes perfect sense. Thank you very much for the help, Giles.