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Binomial, poisson, and gaussian dstributions

bincomp.m Graphical comparison of the binomial, Poisson, and Gaussian distributions. The procedure calls for binomial parameters n , p , determines a reasonable range of evaluation points and plots on the same graph the binomial distribution function, thePoisson distribution function, and the gaussian distribution function with the adjustment called the “continuity correction.”

% BINCOMP file bincomp.m Approx of binomial by Poisson and gaussian % Version of 5/24/96% Gaussian adjusted for "continuity correction" % Plots distribution functions for specified parameters n, pn = input('Enter the parameter n '); p = input('Enter the parameter p ');a = floor(n*p-2*sqrt(n*p)); a = max(a,1); % Prevents zero or negative indicesb = floor(n*p+2*sqrt(n*p)); k = a:b;Fb = cumsum(ibinom(n,p,0:n)); % Binomial distribution function Fp = cumsum(ipoisson(n*p,0:n)); % Poisson distribution functionFg = gaussian(n*p,n*p*(1 - p),k+0.5); % Gaussian distribution function stairs(k,Fb(k+1)) % Plotting detailshold on plot(k,Fp(k+1),'-.',k,Fg,'o')hold off xlabel('t values') % Graph labeling detailsylabel('Distribution function') title('Approximation of Binomial by Poisson and Gaussian')grid legend('Binomial','Poisson','Adjusted Gaussian')disp('See Figure for results')
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poissapp.m Graphical comparison of the Poisson and Gaussian distributions. The procedure calls for a value of the Poisson parameter mu, then calculates and plots the Poissondistribution function, the Gaussian distribution function, and the adjusted Gaussian distribution function.

% POISSAPP file poissapp.m Comparison of Poisson and gaussian % Version of 5/24/96% Plots distribution functions for specified parameter mu mu = input('Enter the parameter mu ');n = floor(1.5*mu); k = floor(mu-2*sqrt(mu)):floor(mu+2*sqrt(mu));FP = cumsum(ipoisson(mu,0:n)); FG = gaussian(mu,mu,k);FC = gaussian(mu,mu,k-0.5); stairs(k,FP(k))hold on plot(k,FG,'-.',k,FC,'o')hold off gridxlabel('t values') ylabel('Distribution function')title('Gaussian Approximation to Poisson Distribution') legend('Poisson','Gaussian','Adjusted Gaussian')disp('See Figure for results')
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Setup for simple random variables

If a simple random variable X is in canonical form, the distribution consists of the coefficients of the indicator funtions (the values of X ) and the probabilities of the corresponding events. If X is in a primitive form other than canonical, the csort operation is applied to the coefficients of the indicator functions and the probabilities of the corresponding events to obtainthe distribution. If Z = g ( X ) and X is in a primitive form, then the value of Z on the event in the partition associated with t i is g ( t i ) . The distribution for Z is obtained by applying csort to the g ( t i ) and the p i . Similarly, if Z = g ( X , Y ) and the joint distribution is available, the value g ( t i , u j ) is associated with P ( X = t i , Y = u j ) . The distribution for Z is obtained by applying csort to the matrix of values and the corresponding matrix of probabilities.

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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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