chapter 2 page 28

The derivation of continuous immediate annuity is unclear. How did the w derived from dw/dt by integrating it comes to -tPx in the first method ?
In the second method of deriving continuous immediate annuity I did not understand how to reverse the order of integration?

 

1. -tpx=-1+tqx it's derivative wrt t is 0 + tpx*mu(x+t)

2. From 1st line integral... 0<s<t<infty
So if you change t: (0, infty) indepemdent to s: (0, infty) independent.
You must change s: (0,t) conditional to t to t: (s, infty) conditional to s.

 

I understood the second method of changing the integral but the first method's derivative thing is still unclear. How is
-tPx =-1 +tqx and could you please explain with meaning of the equations?

 

tpx is the probability of life aged x being alive at age x+1
tqx is the probability of life aged x not being alive at age x+1
Both are mutually exclusive and exhaustive events, so both sum to 1.

And we know tqx(CDF of lifetime) is the integration of tpx*mu(x+t)

 

Yeah then if we integrate from 0 to infinity tpx*mu(x+t) we will get tqx so where has the - 1 gone. Is it the constant of integration? Because if there is no -1 then we can't express it w = - tPx, right?

 

No! -1 came because tpx+tqx=1 as I said before they're mutual exclusive and exhaustive events

And for tqx it is integration from 0 to t

 

So we have to go backwards to understand it like we have to take w=-tPx and then show it's derivative is tPx*mu(x+t) rather than getting w by integrating tpx*mu(x+t)? Because if we do so we have to take limits and what limit to take is uncertain whether 0 to t or 0 to infinity !

 

Both will give same answer but w=-tpx will be easier.

And we know w=-tpx=-1 + integration of that from 0 to t. So its derivative will be 0 + tpx*mu(x+t)

 

Okay thank you Sir !