Appendix a to applied probability: directory of m-functions and m  (Page 22/24)

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Compound demand

The following pattern provides a useful model in many situations. Consider

$D=\sum _{k=0}^{N}{Y}_{k}$

where ${Y}_{0}=0$ , and the class $\left\{{Y}_{k}:1\le k\right\}$ is iid, independent of the counting random variable N . One natural interpretation is to consider N to be the number of customers in a store and Y k the amount purchased by the k th customer. Then D is the total demand of the actual customers. Hence, we call D the compound demand .

gend.m Uses coefficients of the generating functions for N and Y to calculate, in the integer case, the marginal distribution for the compound demand D and the joint distribution for $\left\{N,D\right\}$ .

% GEND file gend.m Marginal and joint dbn for integer compound demand % Version of 5/21/97% Calculates marginal distribution for compound demand D % and joint distribution for {N,D} in the integer case% Do not forget zero coefficients for missing powers % in the generating functions for N, Ydisp('Do not forget zero coefficients for missing powers') gn = input('Enter gen fn COEFFICIENTS for gN ');gy = input('Enter gen fn COEFFICIENTS for gY '); n = length(gn) - 1; % Highest power in gNm = length(gy) - 1; % Highest power in gY P = zeros(n + 1,n*m + 1); % Base for generating Py = 1; % Initialization P(1,1) = gn(1); % First row of P (P(N=0) in the first position)for i = 1:n % Row by row determination of P y = conv(y,gy); % Successive powers of gyP(i+1,1:i*m+1) = y*gn(i+1); % Successive rows of P endPD = sum(P); % Probability for each possible value of D a = find(gn); % Location of nonzero N probabilitiesb = find(PD); % Location of nonzero D probabilities P = P(a,b); % Removal of zero rows and columnsP = rot90(P); % Orientation as on the plane N = 0:n;N = N(a); % N values with positive probabilites PN = gn(a); % Positive N probabilitiesY = 0:m; % All possible values of Y Y = Y(find(gy)); % Y values with positive probabilitiesPY = gy(find(gy)); % Positive Y proabilities D = 0:n*m; % All possible values of DPD = PD(b); % Positive D probabilities D = D(b); % D values with positive probabilitiesgD = [D; PD]'; % Display combinationdisp('Results are in N, PN, Y, PY, D, PD, P') disp('May use jcalc or jcalcf on N, D, P')disp('To view distribution for D, call for gD')

gendf.m function [d,pd] = gendf(gn,gy) is a function version of gend , which allows arbitrary naming of the variables. Calculates the distribution for D , but not the joint distribution for $\left\{N,D\right\}$ .

function [d,pd] = gendf(gn,gy)% GENDF [d,pd] = gendf(gN,gY) Function version of gend.m% Calculates marginal for D in the integer case % Version of 5/21/97% Do not forget zero coefficients for missing powers % in the generating functions for N, Yn = length(gn) - 1; % Highest power in gN m = length(gy) - 1; % Highest power in gYP = zeros(n + 1,n*m + 1); % Base for generating P y = 1; % InitializationP(1,1) = gn(1); % First row of P (P(N=0) in the first position) for i = 1:n % Row by row determination of Py = conv(y,gy); % Successive powers of gy P(i+1,1:i*m+1) = y*gn(i+1); % Successive rows of Pend PD = sum(P); % Probability for each possible value of DD = 0:n*m; % All possible values of D b = find(PD); % Location of nonzero D probabilitiesd = D(b); % D values with positive probabilities pd = PD(b); % Positive D probabilities

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