## Probability distributions based on difference differential equations for pure birth processes

##### Abstract

The main objective of this work is to identify probability distributions emerging by solving
difference- differential equations of a pure birth process given by p~ ( t) = - "-0 Po(t) for n = 0
and p~ (t) = - "-n Pn( t) + "-n,1Pn,I ( t ), for n > 1, The special cases are: Poisson Process
("-n = ,,-), Simple Birth process ("-n = n"-), in Simple Birth process with immigration
("-n = n,,- + v) and the Polya process ["-n = ( 1 + an J"-], 1 + "-at
Four methods have been applied in solving the difference - differential equations are:
(1) the iterative technique
(2) the Laplace Method
(3) the Langranges Method
(4) .the generator Matrix technique.
The results through the four Methods are similar. The means and Variances were obtained by
definition, the pgftechnique and by the method of moments:
The results are:
From the Poisson process, we obtain Poisson distribution with parameter ,,-t both when the
e').!(,,-t r
initial condition is X (0) = 0 and also when X( 0) = no' i.e. Pn (t) = for n = 0, 1,2, .
n!
t
_ e').!("-t)k
P ( ) - for n=ny+k and k=O, 1,2, ....,
n k!
The mean is E [X (t )] = "-t and the variance is Var [X ( t )] = "-t.
From the Simple Birth process we obtain a geometric distribution, Pn(t) = e'At(1 - e'A!r,
when the initial condition is X(0) = 1 and a negative binomial distribution,
Pno+k(t) = (k + kno - IJ x (1 -~,! ).)k X (e'!). )no, k = 0, 1,2, ..., when the i.ni.t.ial condi.t.i.on IS
X(O) = no'
The mean is E(X)= noe')'! and variance noeH!(l- e'A.!).
From the Simple Birth Process with immigration, we obtain a negative binomial distribution
V(1 -At)
no+ - - e ( v) The mean E[X(t)J= \-i-t and the variance Var[X(t)] = no + i eAt(eAt -1).
From the Polya process, we obtain a negative binomial distribution
( J ( J
n +1.+k-1 1 It 0 + 1a ( 1 Jk
p (t) = 0 a X X 1 _ --~
no +k k 1 + Aat (1 + at Y' k=O, 1,2, ...
The mean is At and the variance Yare X (t) ] = At(1 + aAt).
Exceptions
Laplace Method did not work for the Polya process
Generator matrix could not work for the Poisson process.

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