*** Binomial mortality model, survival function for network aging modeling !!!!

Eq 7b seems to have errors? The following revised form is more meaningful. 

INVERSE Function of binomial CDF 

Functions for life span distribution. GG91

page 43, two parameter Gompertz model

page 44, Gompertz-Makeham model

Quadratic form of Gompertz-Makeham model, El Shaarawi et al 1974
  u(t) = M + R exp( at + bt^2)

Logistic equations like the Perks equation, which can be derived from GG91's avalanche-like aging model.

 Weibull model (1951): u(t) = B * t^c

page 46, GG91's generalized bionomial mortality model:
  u(t) = A + (b + c t)^n
GG91 argues that when b >>c, bionomial model lead to R=b^n, G=nc/b
When b<<c,  it is the Weibull model.

Estimation of mortality rate at time $t$, GG91

page 41, Sacher 1956, 1966

  \mu(t) = 1/(2 dt) log( \frac{s(t-dt}{t+dt} )

Other methods discussed are: Cutler and Ederer 1958

GG91 argued that Saccher56 formula is better than CutlerEdere58 method.

Variability in lifespan, GG91 notes

page 31,
There are two extreme positions to explain lifetime variability: heterogeneity hypothesis and stochastic hypothesis.

page 37
GG91 argued that stochastic variation should be considered in addition to environment and genetic factors. GG91 cited Sacher 1977 who tried to quantify stochastic variation from lifespan distribution.
GG91 viewed stochastic variation as 'kinetic' variation.

Heritability of lifespan, GG91, notes

GG91, p33, cite Jacquard, 1982. Heritability coefficient for human lifespan is 0.16.  GG91 interpret this as: Even if the lifespan of both parents exceed the mean lifespan by 20 years, their offspring will gain on average only 0.16x20=3.2 extra years from their genetic 'inheritance'.

Page 34. GG91 then argued that "selection for an increase in life span is ineffective". GG91 then used the heterozygote Aa example to make their point.

page 35,
GG91 cited that heteozygotes often live longer than homozygotes. Bileva et al 1978, 1981, Nekrasov 1981, Shakhbazov 1980.

History for the biology of life span, GG91, reading notes

John Graunt, 1662, first life table for London residents

Leonhard Euler, 1760, general investigations into mortality and the multiplication of the human race

Benjanmin Gompertz, 1825, Gompertz model. (Was Gompertz the first person to use the term "the force of mortality"?)

William Makeham, 1860, Gompertz-Makeham model

Rosset 1979 reviewed the history of biology of lifespan in a Polish article.

mixture models, pdf, CDF, and intensity functions, GG01, GG91, !!!!!!!!!!

Mixture models frequently are applied to the probability density function, not the cumulative functions. But, based on wikipedia entry on "mixture density" , mixture can be defined in both pdf and CDF, and it seems to interchangeable.  In fact, this is quit natural, because mixture is a linear form. Derivative of linear combinations of CDFs lead to linear combinations of pdfs.

GG01 applied binomial formula to the mortality rate functions, not the viability functions. GG91 used the same approach, page 258-261, section 6.6.

What would happen if we apply binomial formula to the survival functions? Interestingly, GG91 page 265-266, section 6.7, applied binomial formula to the failure CDF function, $F(x)$ in GG01. So, proof changed from GG91 to GG01.

The failure CDF for binomial active change of $q$ of $n$ elements is:
 $ F(x) = (1- q exp(-kx))^n $,
which is a pleasantly clean form. 

In contrast, Witten85 applied weights to the reliability (viability) function, though his formulation of the Gompertz model did not use this approach.

The more that I compare GG91 and GG01, the more it looks like linear rules are the same between viability function and intensity functions (mortality rates). However, I have not been able to prove this myself. In fact, I found it otherwise, somehow. 

Reference:

http://en.wikipedia.org/wiki/Mixture_density