Ancora sul processo stocastico nascite-morti

Abstract

Si determinano direttamente le probabilità di stato del processo nascite-morti cumulativo per ricorrenza. Si dà un suggerimento per il calcolo e si confrontano i risultati con quanto già ricavato da D. G. Kendall.

Abstract

In [4] D. G. Kendall left unsolved the equation on the bivariate generating function of the cumulative birth and death process for generalλ(t) andμ(t).

This process is described by means of two state variables:N(t), number of living individuals at timet, M(t) total number of born individuals at timet.

If the generating function method is abandoned and the recurrent equation on state probabilities is reconsidered, a solution is possible. Grouping states carefully in different families, one gets easy equations that, once solved, give formulas (2) (3) (4) (5) (6) (7). These formulas, via recurrency, give probabilities for every state.

In section (3) a method is proposed to make computations easier. This method makes possible a formal approach through the use of formulas (8).

A table can thus be constructed, which, through the sum, row by row, of the individual values, gives the marginal probabilities of theM(t) process.

Forλ(t) ≡λ ₀,μ(t) ≡μ ₀, a comparison is carried out with results from Kendall’s generating function in section 4).