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CEE162/262c Lecture 13: Discrete Stochasticity

CEE162/262c Lecture 13: Discrete Stochasticity. c = 4 c = 3 c = 2 c = 1 c = 0. j = 0 j = 1 j = 2 j = 3 …. j = n. m+1 m. j j+1. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %

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CEE162/262c Lecture 13: Discrete Stochasticity

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  1. CEE162/262c Lecture 13: Discrete Stochasticity CEE162/262c Lecture 13: Discrete stochasticity

  2. CEE162/262c Lecture 13: Discrete stochasticity

  3. CEE162/262c Lecture 13: Discrete stochasticity

  4. CEE162/262c Lecture 13: Discrete stochasticity

  5. CEE162/262c Lecture 13: Discrete stochasticity

  6. CEE162/262c Lecture 13: Discrete stochasticity

  7. c = 4 c = 3 c = 2 c = 1 c = 0 j = 0 j = 1 j = 2 j = 3 …. j = n CEE162/262c Lecture 13: Discrete stochasticity

  8. m+1 m j j+1 CEE162/262c Lecture 13: Discrete stochasticity

  9. CEE162/262c Lecture 13: Discrete stochasticity

  10. CEE162/262c Lecture 13: Discrete stochasticity

  11. CEE162/262c Lecture 13: Discrete stochasticity

  12. CEE162/262c Lecture 13: Discrete stochasticity

  13. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % File: jumps.m % Description: Stochastic test of the binomial distribution. % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% function jumps() nmax = 100; Pc = 0.04; trials = [10,100,1000,10000]; for numtrials=trials k = sum(rand(nmax,numtrials)<Pc); N = hist(k,[0:nmax])/numtrials; kt = [0:nmax]; p = choose(nmax,kt).*Pc.^kt.*(1-Pc).^(nmax-kt); sp = find(numtrials==trials); figure(1); subplot(length(trials),1,sp); plot(kt,N,'k.-',kt,p,'k:','markersize',10); if(sp<length(trials)) set(gca,'xticklabel',[]); end title(sprintf('%d trials',numtrials)); axis([0 20 0 .25]); end function c = choose(n,k) c = factorial(n)./(factorial(n-k).*factorial(k)); CEE162/262c Lecture 13: Discrete stochasticity

  14. CEE162/262c Lecture 13: Discrete stochasticity

  15. Binomial distribution approaches a normal distribution as n infinity. CEE162/262c Lecture 13: Discrete stochasticity

  16. CEE162/262c Lecture 13: Discrete stochasticity

  17. CEE162/262c Lecture 13: Discrete stochasticity

  18. CEE162/262c Lecture 13: Discrete stochasticity

  19. CEE162/262c Lecture 13: Discrete stochasticity

  20. % cranes.m – Stochastic crane population simulation nmax = 5; Pc = 0.04; numtrials = 10000; x0 = 100; r = 0.4; rc = 0.175; eps = 1e-10; kfound = 0; xend = zeros(numtrials,1); for trial=1:numtrials x = x0; cs = 0; for n=1:nmax s = rand()<Pc; cs = cs + s; x = s*(1+rc)*x + (1-s)*(1+r)*x; end ct(trial) = cs; if(kfound==0) xs = x; kfound = 1; else if(min(abs(x-xs))>eps) xs = [xs,x]; kfound = kfound + 1; end end xend(trial) = x; end xs = sort(xs); [Nc,c] = hist(ct,[0:nmax]); [Nx,x] = hist(xend,xs); fprintf('# Cat.\tp(Cat.)\t\tPop.\t\tp(Pop.)\n'); for n=1:length(Nx) fprintf('%d\t\t%.4f\t\t%.0f\t\t%.4f\n',... c(n),Nc(n)/numtrials,x(n),Nx(n)/numtrials); end Output: # Cat. p(Cat.) Pop. p(Pop.) 0 0.8159 317 0.0008 1 0.1694 378 0.0139 2 0.0139 451 0.1694 3 0.0008 537 0.8159 CEE162/262c Lecture 13: Discrete stochasticity

  21. CEE162/262c Lecture 13: Discrete stochasticity

  22. CEE162/262c Lecture 13: Discrete stochasticity

  23. CEE162/262c Lecture 13: Discrete stochasticity

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