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CEE162/262c Lecture 13: Discrete Stochasticity

This lecture explores discrete stochasticity, focusing on the binomial distribution and its properties. A stochastic test is conducted using binomial trials to observe how the distribution approaches a normal distribution as the number of trials increases. The session includes simulations of population dynamics using stochastic models, demonstrating how factors such as probability and initial conditions impact outcomes. Participants can gain insights into the mathematical principles and applications of stochasticity in real-world scenarios, emphasizing data analysis and probabilistic modeling.

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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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