OK, in answer to my own question let us consider the case of the binomial model.
Homogeneous assumption
Assume that all lives of a given age have the same constant chance of dying Q. Variance of the number of deaths D in population of N lives:
\[Var(D) = NPQ \text{ where } P = 1-Q\]
Heterogeneity observed
N lives of the same age are made up of r homogeneous groups. ith group has ni lives that have a chance qi of dying within that year.
\[N = \sum n_i \text{, } Q = (1/N)*\sum n_iq_i \text{, } D = \sum d_i\]
\[E(D) = \sum E(d_i) = \sum n_iq_i = NQ\]
\begin{eqnarray}
Var(D) & = &
\sum Var(d_i)
\\& = &
\sum n_iq_i(1-q_i)
\\& = &
NQ - \sum n_i q_i^2
\\& = &
....
\\& = &
NPQ -\sum n_i (q_i-Q)^2
\\&<&
NPQ
\end{eqnarray}
This shows that heterogeneity in mortality lowers the variance of the number deaths.
P.S. How do I left align the eqnarray environment above?
Last edited by a moderator: Aug 21, 2014