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mgd.m Uses coefficients for the generating function for N and the distribution for simple Y to calculate the distribution for the compound demand.

% MGD file mgd.m Moment generating function for compound demand % Version of 5/19/97% Uses m-functions csort, mgsum disp('Do not forget zeros coefficients for missing')disp('powers in the generating function for N') disp(' ')g = input('Enter COEFFICIENTS for gN '); y = input('Enter VALUES for Y ');p = input('Enter PROBABILITIES for Y '); n = length(g); % Initializationa = 0; b = 1;D = a; PD = g(1);for i = 2:n [a,b]= mgsum(y,a,p,b); D = [D a]; PD = [PD b*g(i)]; [D,PD]= csort(D,PD); endr = find(PD>1e-13); D = D(r); % Values with positive probabilityPD = PD(r); % Corresponding probabilities mD = [D; PD]'; % Display details disp('Values are in row matrix D; probabilities are in PD.')disp('To view the distribution, call for mD.')
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mgdf.m function [d,pd] = mgdf(pn,y,py) is a function version of mgd , which allows arbitrary naming of the variables. The input matrix p n is the coefficient matrix for the counting random variable generating function. Zeros for the missing powers must be included.The matrices y , p y are the actual values and probabilities of the demand random variable.

function [d,pd] = mgdf(pn,y,py)% MGDF [d,pd] = mgdf(pn,y,py) Function version of mgD% Version of 5/19/97 % Uses m-functions mgsum and csort% Do not forget zeros coefficients for missing % powers in the generating function for Nn = length(pn); % Initialization a = 0;b = 1; d = a;pd = pn(1); for i = 2:n[a,b] = mgsum(y,a,py,b);d = [d a];pd = [pd b*pn(i)];[d,pd] = csort(d,pd);end a = find(pd>1e-13); % Location of positive probabilities pd = pd(a); % Positive probabilitiesd = d(a); % D values with positive probability
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randbern.m Let S be the number of successes in a random number N of Bernoulli trials, with probability p of success on each trial. The procedure randbern takes as inputs the probability p of success and the distribution matrices N , P N for the counting random variable N and calculates the joint distribution for { N , S } and the marginal distribution for S .

% RANDBERN file randbern.m Random number of Bernoulli trials % Version of 12/19/96; notation modified 5/20/97% Joint and marginal distributions for a random number of Bernoulli trials % N is the number of trials% S is the number of successes p = input('Enter the probability of success ');N = input('Enter VALUES of N '); PN = input('Enter PROBABILITIES for N ');n = length(N); m = max(N);S = 0:m; P = zeros(n,m+1);for i = 1:n P(i,1:N(i)+1) = PN(i)*ibinom(N(i),p,0:N(i));end PS = sum(P);P = rot90(P); disp('Joint distribution N, S, P, and marginal PS')
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Simulation of markov systems

inventory1.m Calculates the transition matrix for an ( m , M ) inventory policy. At the end of each period, if the stock is less than a reorder point m , stock is replenished to the level M . Demand in each period is an integer valued random variable Y . Input consists of the parameters m , M and the distribution for Y as a simple random variable (or a discrete approximation).

% INVENTORY1 file inventory1.m Generates P for (m,M) inventory policy % Version of 1/27/97% Data for transition probability calculations % for (m,M) inventory policyM = input('Enter value M of maximum stock '); m = input('Enter value m of reorder point ');Y = input('Enter row vector of demand values '); PY = input('Enter demand probabilities ');states = 0:M; ms = length(states);my = length(Y); % Calculations for determining P[y,s] = meshgrid(Y,states);T = max(0,M-y).*(s<m) + max(0,s-y).*(s>= m); P = zeros(ms,ms);for i = 1:ms [a,b]= meshgrid(T(i,:),states); P(i,:) = PY*(a==b)';end disp('Result is in matrix P')
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Questions & Answers

find the 15th term of the geometric sequince whose first is 18 and last term of 387
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20/(×-6^2)
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ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
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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
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Source:  OpenStax, Applied probability. OpenStax CNX. Aug 31, 2009 Download for free at http://cnx.org/content/col10708/1.6
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