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The following pattern provides a useful model in many situations. Consider
where ${Y}_{0}=0$ , and the class $\{{Y}_{k}:1\le k\}$ 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 $\{N,D\}$ .
% 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
$\{N,D\}$ .
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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