# Appendix a to applied probability: directory of m-functions and m  (Page 24/24)

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branchp.m Calculates the transition matrix for a simple branching process with a specified maximum population. Input consists of the maximum population value M and the coefficient matrix for the generating function for the individual propagation random variables Z i . The latter matrix must include zero coefficients for missing powers.

```% BRANCHP file branchp.m Transition P for simple branching process % Version of 7/25/95% 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')```

chainset.m Sets up for simulation of Markov chains. Inputs are the transition matrix P the set of states, and an optional set of target states. The chain generating procedures listed below assume this procedure has been run.

```% CHAINSET file chainset.m Setup for simulating Markov chains % Version of 1/2/96 Revise 7/31/97 for version 4.2 and 5.1P = input('Enter the transition matrix '); ms = length(P(1,:));MS = 1:ms; states = input('Enter the states if not 1:ms ');if isempty(states) states = MS;end disp('States are')disp([MS;states]')PI = input('Enter the long-run probabilities '); F = [zeros(1,ms); cumsum(P')]'; A = F(:,MS);B = F(:,MS+1); e = input('Enter the set of target states ');ne = length(e); E = zeros(1,ne);for i = 1:ne E(i) = MS(e(i)==states);end disp(' ')disp('Call for for appropriate chain generating procedure')```

mchain.m Assumes chainset has been run. Generates trajectory of specified length, with specified initial state.

```% MCHAIN file mchain.m Simulation of Markov chains % Version of 1/2/96 Revised 7/31/97 for version 4.2 and 5.1% Assumes the procedure chainset has been run n = input('Enter the number n of stages ');st = input('Enter the initial state '); if ~isempty(st)s = MS(st==states); elses = 1; endT = zeros(1,n); % Trajectory in state numbers U = rand(1,n);for i = 1:n T(i) = s;s = ((A(s,:)<U(i))&(U(i)<= B(s,:)))*MS'; endN = 0:n-1; tr = [N;states(T)]'; n10 = min(n,11);TR = tr(1:n10,:); f = ones(1,n)/n;[sn,p] = csort(T,f);if isempty(PI) disp(' State Frac')disp([states; p]')else disp(' State Frac PI')disp([states; p; PI]')end disp('To view the first part of the trajectory of states, call for TR')```

arrival.m Assumes chainset has been run. Calculates repeatedly the arrival time to a prescribed set of states.

```% ARRIVAL file arrival.m Arrival time to a set of states % Version of 1/2/96 Revised 7/31/97 for version 4.2 and 5.1% Calculates repeatedly the arrival % time to a prescribed set of states.% Assumes the procedure chainset has been run. r = input('Enter the number of repetitions ');disp('The target state set is:') disp(e)st = input('Enter the initial state '); if ~isempty(st)s1 = MS(st==states); % Initial state number elses1 = 1; endclear T % Trajectory in state numbers (reset) S = zeros(1,r); % Arrival time for each rep (reset)TS = zeros(1,r); % Terminal state number for each rep (reset) for k = 1:rR = zeros(1,ms); % Indicator for target state numbers R(E) = ones(1,ne); % reset for target state numberss = s1; T(1) = s;i = 1; while R(s) ~= 1 % While s is not a target state numberu = rand(1,1); s = ((A(s,:)<u)&(u<= B(s,:)))*MS'; i = i+1;T(i) = s; endS(k) = i-1; % i is the number of stages; i-1 is time TS(k) = T(i);end [ts,ft]= csort(TS,ones(1,r)); % ts = terminal state numbers ft = frequencies fts = ft/r; % Relative frequency of each ts[a,at] = csort(TS,S); % at = arrival time for each tsw = at./ft; % Average arrival time for each ts RES = [states(ts); fts; w]'; disp(' ')if r == 1 disp(['The arrival time is ',int2str(i-1),]) disp(['The state reached is ',num2str(states(ts)),]) N = 0:i-1;TR = [N;states(T)]';disp('To view the trajectory of states, call for TR') elsedisp(['The result of ',int2str(r),' repetitions is:'])disp('Term state Rel Freq Av time') disp(RES)disp(' ') [t,f]= csort(S,ones(1,r)); % t = arrival times f = frequencies p = f/r; % Relative frequency of each tdbn = [t; p]';AV = dot(t,p); SD = sqrt(dot(t.^2,p) - AV^2);MN = min(t); MX = max(t);disp(['The average arrival time is ',num2str(AV),])disp(['The standard deviation is ',num2str(SD),])disp(['The minimum arrival time is ',int2str(MN),])disp(['The maximum arrival time is ',int2str(MX),])disp('To view the distribution of arrival times, call for dbn') disp('To plot the arrival time distribution, call for plotdbn')end```

recurrence.m Assumes chainset has been run. Calculates repeatedly the recurrence time to a prescribed set of states, if initial state is in the set; otherwise calculatesthe arrival time.

```% RECURRENCE file recurrence.m Recurrence time to a set of states % Version of 1/2/96 Revised 7/31/97 for version 4.2 and 5.1% Calculates repeatedly the recurrence time % to a prescribed set of states, if initial% state is in the set; otherwise arrival time. % Assumes the procedure chainset has been run.r = input('Enter the number of repititions '); disp('The target state set is:')disp(e) st = input('Enter the initial state ');if ~isempty(st) s1 = MS(st==states); % Initial state numberelse s1 = 1;end clear T % Trajectory in state numbers (reset)S = zeros(1,r); % Recurrence time for each rep (reset) TS = zeros(1,r); % Terminal state number for each rep (reset)for k = 1:r R = zeros(1,ms); % Indicator for target state numbersR(E) = ones(1,ne); % reset for target state numbers s = s1;T(1) = s; i = 1;if R(s) == 1 u = rand(1,1);s = ((A(s,:)<u)&(u<= B(s,:)))*MS'; i = i+1;T(i) = s; endwhile R(s) ~= 1 % While s is not a target state number u = rand(1,1);s = ((A(s,:)<u)&(u<= B(s,:)))*MS'; i = i+1;T(i) = s; endS(k) = i-1; % i is the number of stages; i-1 is time TS(k) = T(i);end [ts,ft]= csort(TS,ones(1,r)); % ts = terminal state numbers ft = frequencies fts = ft/r; % Relative frequency of each ts[a,tt] = csort(TS,S); % tt = total time for each tsw = tt./ft; % Average time for each ts RES = [states(ts); fts; w]'; disp(' ')if r == 1 disp(['The recurrence time is ',int2str(i-1),]) disp(['The state reached is ',num2str(states(ts)),]) N = 0:i-1;TR = [N;states(T)]';disp('To view the trajectory of state numbers, call for TR') elsedisp(['The result of ',int2str(r),' repetitions is:'])disp('Term state Rel Freq Av time') disp(RES)disp(' ') [t,f]= csort(S,ones(1,r)); % t = recurrence times f = frequencies p = f/r; % Relative frequency of each tdbn = [t; p]';AV = dot(t,p); SD = sqrt(dot(t.^2,p) - AV^2);MN = min(t); MX = max(t); disp(['The average recurrence time is ',num2str(AV),]) disp(['The standard deviation is ',num2str(SD),]) disp(['The minimum recurrence time is ',int2str(MN),]) disp(['The maximum recurrence time is ',int2str(MX),]) disp('To view the distribution of recurrence times, call for dbn')disp('To plot the recurrence time distribution, call for plotdbn') end```

kvis.m Assumes chainset has been run. Calculates repeatedly the time to complete visits to a specified k of the states in a prescribed set.

```% KVIS file kvis.m Time to complete k visits to a set of states % Version of 1/2/96 Revised 7/31/97 for version 4.2 and 5.1% Calculates repeatedly the time to complete % visits to k of the states in a prescribed set.% Default is visit to all the target states. % Assumes the procedure chainset has been run.r = input('Enter the number of repetitions '); disp('The target state set is:')disp(e) ks = input('Enter the number of target states to visit ');if isempty(ks) ks = ne;end if ks>ne ks = ne;end st = input('Enter the initial state ');if ~isempty(st) s1 = MS(st==states); % Initial state numberelse s1 = 1;end disp(' ')clear T % Trajectory in state numbers (reset) R0 = zeros(1,ms); % Indicator for target state numbersR0(E) = ones(1,ne); % reset S = zeros(1,r); % Terminal transitions for each rep (reset)for k = 1:r R = R0;s = s1; if R(s) == 1R(s) = 0; endi = 1; T(1) = s;while sum(R)>ne - ks u = rand(1,1);s = ((A(s,:)<u)&(u<= B(s,:)))*MS'; if R(s) == 1R(s) = 0; endi = i+1; T(i) = s;end S(k) = i-1;end if r == 1disp(['The time for completion is ',int2str(i-1),])N = 0:i-1; TR = [N;states(T)]'; disp('To view the trajectory of states, call for TR')else [t,f]= csort(S,ones(1,r)); p = f/r;D = [t;f]';AV = dot(t,p); SD = sqrt(dot(t.^2,p) - AV^2);MN = min(t); MX = max(t);disp(['The average completion time is ',num2str(AV),])disp(['The standard deviation is ',num2str(SD),])disp(['The minimum completion time is ',int2str(MN),])disp(['The maximum completion time is ',int2str(MX),])disp(' ') disp('To view a detailed count, call for D.')disp('The first column shows the various completion times;') disp('the second column shows the numbers of trials yielding those times')end```

plotdbn Used after m-procedures arrival or recurrence to plot arrival or recurrence time distribution.

```% PLOTDBN file plotdbn.m % Version of 1/23/98% Plot arrival or recurrence time dbn % Use after procedures arrival or recurrence% to plot arrival or recurrence time distribution plot(t,p,'-',t,p,'+')grid title('Time Distribution')xlabel('Time in number of transitions') ylabel('Relative frequency')```

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A fair die is tossed 180 times. Find the probability P that the face 6 will appear between 29 and 32 times inclusive  By Sam Luong    By   By 