User:Nick Mulgan/sandbox

The maxSPRT is not agreed as the most efficient extension of SPRT to composite hypotheses median = $$I_{\frac{1}{2}}^{[-1]}(\alpha,\beta)(c-a) + a $$ where $$I_{\frac{1}{2}}^{[-1]}(\alpha,\beta)$$ is the 50th percentile of the regularised incomplete beta function. $$\approx a + (c-a) \frac{\alpha-1/3}{\alpha+\beta - 2/3} = \frac{a + 6 b + c}{8}$$ $$\approx \frac{7 a - 6 b - c}{7 a -12 b + 5 c}$$

It retains the algebraic properties power associativity, meaning that if $$s$$ is a sedenion, $$s^n s^m = s^{n + m}$$, and flexibility, meaning that for sedenions $$s$$ and $$t$$, $$(st)s = s(ts)$$, but loses

Mathematical formula
The equation reads:
 * $$\frac{\partial n}{\partial t} + \frac{\partial n}{\partial a} = - m(a)n(t,a) $$

where the population density n(t,a), the number of individuals of age in the range $$[a,a+da]$$ at time t, divided by da, is a function of age a and time t, and m(a) is the death function. (Note that n is a density per age, not per volume or area.)

The boundary conditions are:
 * 1) the birth equation:
 * $$ n(t,0)= \int_\alpha^\beta b (a)n(t,a) \, da$$
 * where $$\alpha$$ and $$\beta$$ are the minimum and maximum fertile ages, $$0\le\alpha\le\beta<\infty$$, $$b(a)$$ is the birth rate, the number of births born in time $$dt$$ per individual of age in the range $$[a,a+da]$$,
 * 2) the initial age structure
 * $$ n(0,a)= f(a) $$, with f known.