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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