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-#
-# $Id: prob2.dem,v 1.9 2006/06/14 03:24:09 sfeam Exp $
-#
-# Demo Statistical Approximations version 1.1
-#
-# Copyright (c) 1991, Jos van der Woude, jvdwoude@hut.nl
-
-# History:
-#    -- --- 1991 Jos van der Woude:  1st version
-#    06 Jun 2006 Dan Sebald:  Added plot methods for better visual effect.
-
-print ""
-print ""
-print ""
-print ""
-print ""
-print ""
-print "                        Statistical Approximations, version 1.1"
-print ""
-print "        Copyright (c) 1991, 1992, Jos van de Woude, jvdwoude@hut.nl"
-print ""
-print ""
-print ""
-print ""
-print ""
-print ""
-print ""
-print ""
-print ""
-print ""
-print ""
-print "     NOTE: contains 10 plots and consequently takes some time to run"
-print "                      Press Ctrl-C to exit right now"
-print ""
-pause -1 "                      Press Return to start demo ..."
-
-load "stat.inc"
-rnd(x) = floor(x+0.5)
-r_xmin = -1
-r_sigma = 4.0
-
-# Binomial PDF using normal approximation
-n = 25; p = 0.15
-mu = n * p
-sigma = sqrt(n * p * (1.0 - p))
-xmin = floor(mu - r_sigma * sigma)
-xmin = xmin < r_xmin ? r_xmin : xmin
-xmax = ceil(mu + r_sigma * sigma)
-ymax = 1.1 * binom(floor((n+1)*p), n, p) #mode of binomial PDF used
-set key box
-unset zeroaxis
-set xrange [xmin - 1 : xmax + 1]
-set yrange [0 : ymax]
-set xlabel "k, x ->"
-set ylabel "probability density ->"
-set ytics 0, ymax / 10.0, ymax
-set format x "%2.0f"
-set format y "%3.2f"
-set sample 200
-set title "binomial PDF using normal approximation"
-set arrow from mu, 0 to mu, normal(mu, mu, sigma) nohead
-set arrow from mu, normal(mu + sigma, mu, sigma) \
-          to mu + sigma, normal(mu + sigma, mu, sigma) nohead
-set label "mu" at mu + 0.5, ymax / 10
-set label "sigma" at mu + 0.5 + sigma, normal(mu + sigma, mu, sigma)
-plot binom(rnd(x), n, p) with histeps, normal(x, mu, sigma)
-pause -1 "Hit return to continue"
-unset arrow
-unset label
-
-# Binomial PDF using poisson approximation
-n = 50; p = 0.1
-mu = n * p
-sigma = sqrt(mu)
-xmin = floor(mu - r_sigma * sigma)
-xmin = xmin < r_xmin ? r_xmin : xmin
-xmax = ceil(mu + r_sigma * sigma)
-ymax = 1.1 * binom(floor((n+1)*p), n, p) #mode of binomial PDF used
-set key box
-unset zeroaxis
-set xrange [xmin - 1 : xmax + 1]
-set yrange [0 : ymax]
-set xlabel "k ->"
-set ylabel "probability density ->"
-set ytics 0, ymax / 10.0, ymax
-set format x "%2.0f"
-set format y "%3.2f"
-set sample (xmax - xmin + 3)
-set title "binomial PDF using poisson approximation"
-set arrow from mu, 0 to mu, normal(mu, mu, sigma) nohead
-set arrow from mu, normal(mu + sigma, mu, sigma) \
-          to mu + sigma, normal(mu + sigma, mu, sigma) nohead
-set label "mu" at mu + 0.5, ymax / 10
-set label "sigma" at mu + 0.5 + sigma, normal(mu + sigma, mu, sigma)
-plot binom(x, n, p) with histeps, poisson(x, mu) with histeps
-pause -1 "Hit return to continue"
-unset arrow
-unset label
-
-# Geometric PDF using gamma approximation
-p = 0.3
-mu = (1.0 - p) / p
-sigma = sqrt(mu / p)
-lambda = p
-rho = 1.0 - p
-xmin = floor(mu - r_sigma * sigma)
-xmin = xmin < r_xmin ? r_xmin : xmin
-xmax = ceil(mu + r_sigma * sigma)
-ymax = 1.1 * p
-set key box
-unset zeroaxis
-set xrange [xmin - 1 : xmax + 1]
-set yrange [0 : ymax]
-set xlabel "k, x ->"
-set ylabel "probability density ->"
-set ytics 0, ymax / 10.0, ymax
-set format x "%2.0f"
-set format y "%3.2f"
-set sample 200
-set title "geometric PDF using gamma approximation"
-set arrow from mu, 0 to mu, gmm(mu, rho, lambda) nohead
-set arrow from mu, gmm(mu + sigma, rho, lambda) \
-          to mu + sigma, gmm(mu + sigma, rho, lambda) nohead
-set label "mu" at mu + 0.5, ymax / 10
-set label "sigma" at mu + 0.5 + sigma, gmm(mu + sigma, rho, lambda)
-plot geometric(rnd(x),p) with histeps, gmm(x, rho, lambda)
-pause -1 "Hit return to continue"
-unset arrow
-unset label
-
-# Geometric PDF using normal approximation
-p = 0.3
-mu = (1.0 - p) / p
-sigma = sqrt(mu / p)
-xmin = floor(mu - r_sigma * sigma)
-xmin = xmin < r_xmin ? r_xmin : xmin
-xmax = ceil(mu + r_sigma * sigma)
-ymax = 1.1 * p
-set key box
-unset zeroaxis
-set xrange [xmin - 1 : xmax + 1]
-set yrange [0 : ymax]
-set xlabel "k, x ->"
-set ylabel "probability density ->"
-set ytics 0, ymax / 10.0, ymax
-set format x "%2.0f"
-set format y "%3.2f"
-set sample 200
-set title "geometric PDF using normal approximation"
-set arrow from mu, 0 to mu, normal(mu, mu, sigma) nohead
-set arrow from mu, normal(mu + sigma, mu, sigma) \
-          to mu + sigma, normal(mu + sigma, mu, sigma) nohead
-set label "mu" at mu + 0.5, ymax / 10
-set label "sigma" at mu + 0.5 + sigma, normal(mu + sigma, mu, sigma)
-plot geometric(rnd(x),p) with histeps, normal(x, mu, sigma)
-pause -1 "Hit return to continue"
-unset arrow
-unset label
-
-# Hypergeometric PDF using binomial approximation
-nn = 75; mm = 25; n = 10
-p = real(mm) / nn
-mu = n * p
-sigma = sqrt(real(nn - n) / (nn - 1.0) * n * p * (1.0 - p))
-xmin = floor(mu - r_sigma * sigma)
-xmin = xmin < r_xmin ? r_xmin : xmin
-xmax = ceil(mu + r_sigma * sigma)
-ymax = 1.1 * hypgeo(floor(mu), nn, mm, n) #mode of binom PDF used
-set key box
-unset zeroaxis
-set xrange [xmin - 1 : xmax + 1]
-set yrange [0 : ymax]
-set xlabel "k ->"
-set ylabel "probability density ->"
-set ytics 0, ymax / 10.0, ymax
-set format x "%2.0f"
-set format y "%3.2f"
-set sample (xmax - xmin + 3)
-set title "hypergeometric PDF using binomial approximation"
-set arrow from mu, 0 to mu, binom(floor(mu), n, p) nohead
-set arrow from mu, binom(floor(mu + sigma), n, p) \
-          to mu + sigma, binom(floor(mu + sigma), n, p) nohead
-set label "mu" at mu + 0.5, ymax / 10
-set label "sigma" at mu + 0.5 + sigma, binom(floor(mu + sigma), n, p)
-plot hypgeo(x, nn, mm, n) with histeps, binom(x, n, p) with histeps
-pause -1 "Hit return to continue"
-unset arrow
-unset label
-
-# Hypergeometric PDF using normal approximation
-nn = 75; mm = 25; n = 10
-p = real(mm) / nn
-mu = n * p
-sigma = sqrt(real(nn - n) / (nn - 1.0) * n * p * (1.0 - p))
-xmin = floor(mu - r_sigma * sigma)
-xmin = xmin < r_xmin ? r_xmin : xmin
-xmax = ceil(mu + r_sigma * sigma)
-ymax = 1.1 * hypgeo(floor(mu), nn, mm, n) #mode of binom PDF used
-set key box
-unset zeroaxis
-set xrange [xmin - 1 : xmax + 1]
-set yrange [0 : ymax]
-set xlabel "k, x ->"
-set ylabel "probability density ->"
-set ytics 0, ymax / 10.0, ymax
-set format x "%2.0f"
-set format y "%3.2f"
-set sample 200
-set title "hypergeometric PDF using normal approximation"
-set arrow from mu, 0 to mu, normal(mu, mu, sigma) nohead
-set arrow from mu, normal(mu + sigma, mu, sigma) \
-          to mu + sigma, normal(mu + sigma, mu, sigma) nohead
-set label "mu" at mu + 0.5, ymax / 10
-set label "sigma" at mu + 0.5 + sigma, normal(mu + sigma, mu, sigma)
-plot hypgeo(rnd(x), nn, mm, n) with histeps, normal(x, mu, sigma)
-pause -1 "Hit return to continue"
-unset arrow
-unset label
-
-# Negative binomial PDF using gamma approximation
-r = 8; p = 0.6
-mu = r * (1.0 - p) / p
-sigma = sqrt(mu / p)
-lambda = p
-rho = r * (1.0 - p)
-xmin = floor(mu - r_sigma * sigma)
-xmin = xmin < r_xmin ? r_xmin : xmin
-xmax = ceil(mu + r_sigma * sigma)
-ymax = 1.1 * gmm((rho - 1) / lambda, rho, lambda) #mode of gamma PDF used
-set key box
-unset zeroaxis
-set xrange [xmin - 1 : xmax + 1]
-set yrange [0 : ymax]
-set xlabel "k, x ->"
-set ylabel "probability density ->"
-set ytics 0, ymax / 10.0, ymax
-set format x "%2.0f"
-set format y "%3.2f"
-set sample 200
-set title "negative binomial PDF using gamma approximation"
-set arrow from mu, 0 to mu, gmm(mu, rho, lambda) nohead
-set arrow from mu, gmm(mu + sigma, rho, lambda) \
-          to mu + sigma, gmm(mu + sigma, rho, lambda) nohead
-set label "mu" at mu + 0.5, ymax / 10
-set label "sigma" at mu + 0.5 + sigma, gmm(mu + sigma, rho, lambda)
-plot negbin(rnd(x), r, p) with histeps, gmm(x, rho, lambda)
-pause -1 "Hit return to continue"
-unset arrow
-unset label
-
-# Negative binomial PDF using normal approximation
-r = 8; p = 0.4
-mu = r * (1.0 - p) / p
-sigma = sqrt(mu / p)
-xmin = floor(mu - r_sigma * sigma)
-xmin = xmin < r_xmin ? r_xmin : xmin
-xmax = ceil(mu + r_sigma * sigma)
-ymax = 1.1 * negbin(floor((r-1)*(1-p)/p), r, p) #mode of gamma PDF used
-set key box
-unset zeroaxis
-set xrange [xmin - 1 : xmax + 1]
-set yrange [0 : ymax]
-set xlabel "k, x ->"
-set ylabel "probability density ->"
-set ytics 0, ymax / 10.0, ymax
-set format x "%2.0f"
-set format y "%3.2f"
-set sample 200
-set title "negative binomial PDF using normal approximation"
-set arrow from mu, 0 to mu, normal(mu, mu, sigma) nohead
-set arrow from mu, normal(mu + sigma, mu, sigma) \
-          to mu + sigma, normal(mu + sigma, mu, sigma) nohead
-set label "mu" at mu + 0.5, ymax / 10
-set label "sigma" at mu + 0.5 + sigma, normal(mu + sigma, mu, sigma)
-plot negbin(rnd(x), r, p) with histeps, normal(x, mu, sigma)
-pause -1 "Hit return to continue"
-unset arrow
-unset label
-
-# Normal PDF using logistic approximation
-mu = 1.0; sigma = 1.5
-a = mu
-lambda = pi / (sqrt(3.0) * sigma)
-xmin = mu - r_sigma * sigma
-xmax = mu + r_sigma * sigma
-ymax = 1.1 * logistic(mu, a, lambda) #mode of logistic PDF used
-set key box
-unset zeroaxis
-set xrange [xmin: xmax]
-set yrange [0 : ymax]
-set xlabel "x ->"
-set ylabel "probability density ->"
-set ytics 0, ymax / 10.0, ymax
-set format x "%.1f"
-set format y "%.2f"
-set sample 200
-set title "normal PDF using logistic approximation"
-set arrow from mu,0 to mu, normal(mu, mu, sigma) nohead
-set arrow from mu, normal(mu + sigma, mu, sigma) \
-          to mu + sigma, normal(mu + sigma, mu, sigma) nohead
-set label "mu" at mu + 0.5, ymax / 10
-set label "sigma" at mu + 0.5 + sigma, normal(mu + sigma, mu, sigma)
-plot logistic(x, a, lambda), normal(x, mu, sigma)
-pause -1 "Hit return to continue"
-unset arrow
-unset label
-
-# Poisson PDF using normal approximation
-mu = 5.0
-sigma = sqrt(mu)
-xmin = floor(mu - r_sigma * sigma)
-xmin = xmin < r_xmin ? r_xmin : xmin
-xmax = ceil(mu + r_sigma * sigma)
-ymax = 1.1 * poisson(mu, mu) #mode of poisson PDF used
-set key box
-unset zeroaxis
-set xrange [xmin - 1 : xmax + 1]
-set yrange [0 : ymax]
-set xlabel "k, x ->"
-set ylabel "probability density ->"
-set ytics 0, ymax / 10.0, ymax
-set format x "%2.0f"
-set format y "%3.2f"
-set sample 200
-set title "poisson PDF using normal approximation"
-set arrow from mu, 0 to mu, normal(mu, mu, sigma) nohead
-set arrow from mu, normal(mu + sigma, mu, sigma) \
-          to mu + sigma, normal(mu + sigma, mu, sigma) nohead
-set label "mu" at mu + 0.5, ymax / 10
-set label "sigma" at mu + 0.5 + sigma, normal(mu + sigma, mu, sigma)
-plot poisson(rnd(x), mu) with histeps, normal(x, mu, sigma)
-pause -1 "Hit return to continue"
-reset