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The element on row i and column j is the probability . Thus, the elements on the i th row constitute the conditional distribution for , given . The transition matrix thus has the property that each row sums to one . Such a matrix is called a stochastic matrix . We return to the examples. From the propositions on transition probabilities, it isapparent that each is Markov. Since the function g is the same for all n and the driving random variables corresponding to the Y i form an iid class, the sequences must be homogeneous. We may utilize part (b) of the propositionsto obtain the one-step transition probabilities.
, so that g n is invariant with n . Since is iid,
and . If is iid, then
We thus have
With the aid of moment generating functions, one may determine distributions for
These calculations are implemented in an m-procedure called branchp . We simply need the distribution for the iid .
% file branchp.m
% Calculates transition matrix for a simple branching
% process with specified maximum population.disp('Do not forget zero probabilities for missing values of Z')
PZ = input('Enter PROBABILITIES for individuals ');M = input('Enter maximum allowable population ');
mz = length(PZ) - 1;EZ = dot(0:mz,PZ);
disp(['The average individual propagation is ',num2str(EZ),])
P = zeros(M+1,M+1);Z = zeros(M,M*mz+1);
k = 0:M*mz;a = min(M,k);
z = 1;P(1,1) = 1;
for i = 1:M % Operation similar to genDz = conv(PZ,z);
Z(i,1:i*mz+1) = z;[t,p] = csort(a,Z(i,:));P(i+1,:) = p;
enddisp('The transition matrix is P')
disp('To study the evolution of the process, call for branchdbn')PZ = 0.01*[15 45 25 10 5]; % Probability distribution for individuals
branchp % Call for procedureDo not forget zero probabilities for missing values of Z
Enter PROBABILITIES for individuals PZEnter maximum allowable population 10
The average individual propagation is 1.45The transition matrix is P
To study the evolution of the process, call for branchdbndisp(P) % Optional display of generated P
Columns 1 through 71.0000 0 0 0 0 0 0
0.1500 0.4500 0.2500 0.1000 0.0500 0 00.0225 0.1350 0.2775 0.2550 0.1675 0.0950 0.0350
0.0034 0.0304 0.1080 0.1991 0.2239 0.1879 0.12930.0005 0.0061 0.0307 0.0864 0.1534 0.1910 0.1852
0.0001 0.0011 0.0075 0.0284 0.0702 0.1227 0.16230.0000 0.0002 0.0017 0.0079 0.0253 0.0579 0.1003
0.0000 0.0000 0.0003 0.0020 0.0078 0.0222 0.04830.0000 0.0000 0.0001 0.0005 0.0021 0.0074 0.0194
0.0000 0.0000 0.0000 0.0001 0.0005 0.0022 0.00680.0000 0.0000 0.0000 0.0000 0.0001 0.0006 0.0022
Columns 8 through 110 0 0 0
0 0 0 00.0100 0.0025 0 0
0.0705 0.0315 0.0119 0.00430.1481 0.0987 0.0559 0.0440
0.1730 0.1545 0.1179 0.16250.1381 0.1574 0.1528 0.3585
0.0832 0.1179 0.1412 0.57710.0406 0.0698 0.1010 0.7591
0.0169 0.0345 0.0590 0.87990.0062 0.0147 0.0294 0.9468
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