# # $Id: prob2.demo 3.38.2.6 1992/11/14 02:25:21 woo Exp $ # # Demo Statistical Approximations version 1.1 # # Permission granted to distribute freely for non-commercial purposes only # # Copyright (c) 1991, Jos van der Woude, jvdwoude@hut.nl pause 0 "" pause 0 "" pause 0 "" pause 0 "" pause 0 "" pause 0 "" pause 0 " Statistical Approximations, version 1.1" pause 0 "" pause 0 " Copyright (c) 1991, 1992, Jos van de Woude, jvdwoude@hut.nl" pause 0 "" pause 0 "" pause 0 " Permission granted to distribute freely for non-commercial purposes only" pause 0 "" pause 0 "" pause 0 "" pause 0 "" pause 0 "" pause 0 "" pause 0 "" pause 0 "" pause 0 "" pause 0 " NOTE: contains 10 plots and consequently takes some time to run" pause 0 " Press Ctrl-C to exit right now" pause 0 "" pause -1 " Press Return to start demo ..." save set "defaults.ini" load "stat.inc" # Binomial PDF using normal approximation n = 25; p = 0.15 mu = n * p sigma = sqrt(n * p * (1.0 - p)) xmin = floor(mu - 4.0 * sigma) xmin = xmin < 0 ? 0 : xmin xmax = ceil(mu + 4.0 * sigma) ymax = 1.1 * binom(mu) #mode of binomial PDF used xinc = ceil((xmax - xmin) / 10) xinc = xinc > 1 ? xinc : 1 set key set nozeroaxis set xrange [xmin - 1 : xmax + 1] set yrange [0 : ymax] set xlabel "k, x ->" set ylabel "probability density ->" set xtics xmin + 0.499, ceil(sigma), xmax 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 + 0.5, 0 to mu + 0.5, normal(mu) nohead set arrow from mu + 0.5, normal(mu + sigma) \ to mu + 0.5 + sigma, normal(mu + sigma) nohead set label "mu" at mu + 1, ymax / 10 set label "sigma" at mu + 1 + sigma, normal(mu + sigma) plot binom(x), normal(x - 0.5) pause -1 "Hit return to continue" set noarrow set nolabel # Binomial PDF using poisson approximation n = 50; p = 0.1 mu = n * p sigma = sqrt(mu) xmin = floor(mu - 4.0 * sigma) xmin = xmin < 0 ? 0 : xmin xmax = ceil(mu + 4.0 * sigma) ymax = 1.1 * binom(mu) #mode of binomial PDF used xinc = ceil((xmax - xmin) / 10) xinc = xinc > 1 ? xinc : 1 set key set nozeroaxis set xrange [xmin - 1 : xmax + 1] set yrange [0 : ymax] set xlabel "k ->" set ylabel "probability density ->" set xtics xmin + 0.499, ceil(sigma), xmax 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 poisson approximation" set arrow from mu + 0.5, 0 to mu + 0.5, normal(mu) nohead set arrow from mu + 0.5, normal(mu + sigma) \ to mu + 0.5 + sigma, normal(mu + sigma) nohead set label "mu" at mu + 1, ymax / 10 set label "sigma" at mu + 1 + sigma, normal(mu + sigma) plot binom(x), poisson(x) pause -1 "Hit return to continue" set noarrow set nolabel # 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 - 4.0 * sigma) xmin = xmin < 0 ? 0 : xmin xmax = ceil(mu + 4.0 * sigma) xinc = ceil((xmax - xmin) / 10) xinc = xinc > 1 ? xinc : 1 ymax = 1.1 * p set key set nozeroaxis set xrange [xmin - 1 : xmax + 1] set yrange [0 : ymax] set xlabel "k, x ->" set ylabel "probability density ->" set xtics xmin + 0.499, ceil((xmax - xmin)/ 10.0), xmax 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 + 0.5, 0 to mu + 0.5, g(mu) nohead set arrow from mu + 0.5, g(mu + sigma) \ to mu + 0.5 + sigma, g(mu + sigma) nohead set label "mu" at mu + 1, ymax / 10 set label "sigma" at mu + 1 + sigma, g(mu + sigma) plot geometric(x), g(x - 0.5) pause -1 "Hit return to continue" set noarrow set nolabel # Geometric PDF using normal approximation p = 0.3 mu = (1.0 - p) / p sigma = sqrt(mu / p) xmin = floor(mu - 4.0 * sigma) xmin = xmin < 0 ? 0 : xmin xmax = ceil(mu + 4.0 * sigma) xinc = ceil((xmax - xmin) / 10) xinc = xinc > 1 ? xinc : 1 ymax = 1.1 * p set key set nozeroaxis set xrange [xmin - 1 : xmax + 1] set yrange [0 : ymax] set xlabel "k, x ->" set ylabel "probability density ->" set xtics xmin + 0.499, ceil((xmax - xmin)/ 10.0), xmax 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 + 0.5, 0 to mu + 0.5, normal(mu) nohead set arrow from mu + 0.5, normal(mu + sigma) \ to mu + 0.5 + sigma, normal(mu + sigma) nohead set label "mu" at mu + 1, ymax / 10 set label "sigma" at mu + 1 + sigma, normal(mu + sigma) plot geometric(x), normal(x - 0.5) pause -1 "Hit return to continue" set noarrow set nolabel # 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 - 4.0 * sigma) xmin = xmin < 0 ? 0 : xmin xmax = ceil(mu + 4.0 * sigma) xinc = ceil((xmax - xmin) / 10) xinc = xinc > 1 ? xinc : 1 ymax = 1.1 * hypgeo(mu) #mode of binom PDF used set key set nozeroaxis set xrange [xmin - 1 : xmax + 1] set yrange [0 : ymax] set xlabel "k ->" set ylabel "probability density ->" set xtics xmin, xinc, xmax 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 binomial approximation" set arrow from mu + 0.5, 0 to mu + 0.5, binom(mu) nohead set arrow from mu + 0.5, binom(mu + sigma) \ to mu + 0.5 + sigma, binom(mu + sigma) nohead set label "mu" at mu + 1, ymax / 10 set label "sigma" at mu + 1 + sigma, binom(mu + sigma) plot hypgeo(x), binom(x) pause -1 "Hit return to continue" set noarrow set nolabel # 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 - 4.0 * sigma) xmin = xmin < 0 ? 0 : xmin xmax = ceil(mu + 4.0 * sigma) xinc = ceil((xmax - xmin) / 10) xinc = xinc > 1 ? xinc : 1 ymax = 1.1 * hypgeo(mu) #mode of binom PDF used set key set nozeroaxis set xrange [xmin - 1 : xmax + 1] set yrange [0 : ymax] set xlabel "k, x ->" set ylabel "probability density ->" set xtics xmin, xinc, xmax 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 + 0.5, 0 to mu + 0.5, normal(mu) nohead set arrow from mu + 0.5, normal(mu + sigma) \ to mu + 0.5 + sigma, normal(mu + sigma) nohead set label "mu" at mu + 1, ymax / 10 set label "sigma" at mu + 1 + sigma, normal(mu + sigma) plot hypgeo(x), normal(x - 0.5) pause -1 "Hit return to continue" set noarrow set nolabel # 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 = int(mu - 4.0 * sigma) xmin = xmin < 0 ? 0 : xmin xmax = int(mu + 4.0 * sigma) xinc = ceil((xmax - xmin) / 10) xinc = xinc > 1 ? xinc : 1 ymax = 1.1 * g((rho - 1) / lambda) #mode of gamma PDF used set key set nozeroaxis set xrange [xmin - 1 : xmax + 1] set yrange [0 : ymax] set xlabel "k, x ->" set ylabel "probability density ->" set xtics xmin + 0.499, ceil((xmax - xmin)/ 10.0), xmax 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 + 0.5, 0 to mu + 0.5, g(mu) nohead set arrow from mu + 0.5, g(mu + sigma) \ to mu + 0.5 + sigma, g(mu + sigma) nohead set label "mu" at mu + 1, ymax / 10 set label "sigma" at mu + 1 + sigma, g(mu + sigma) plot negbin(x), g(x - 0.5) pause -1 "Hit return to continue" set noarrow set nolabel # Negative binomial PDF using normal approximation r = 8; p = 0.4 mu = r * (1.0 - p) / p sigma = sqrt(mu / p) xmin = floor(mu - 4.0 * sigma) xmin = xmin < 0 ? 0 : xmin xmax = ceil(mu + 4.0 * sigma) xinc = ceil((xmax - xmin) / 10) xinc = xinc > 1 ? xinc : 1 ymax = 1.1 * negbin(mu - 1/p) #mode of gamma PDF used set key set nozeroaxis set xrange [xmin - 1 : xmax + 1] set yrange [0 : ymax] set xlabel "k, x ->" set ylabel "probability density ->" set xtics xmin + 0.499, ceil((xmax - xmin)/ 10.0), xmax 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 + 0.5, 0 to mu + 0.5, normal(mu) nohead set arrow from mu + 0.5, normal(mu + sigma) \ to mu + 0.5 + sigma, normal(mu + sigma) nohead set label "mu" at mu + 1, ymax / 10 set label "sigma" at mu + 1 + sigma, normal(mu + sigma) plot negbin(x), normal(x - 0.5) pause -1 "Hit return to continue" set noarrow set nolabel # Normal PDF using logistic approximation mu = 1.0; sigma = 1.5 a = mu lambda = pi / (sqrt(3.0) * sigma) xmin = mu - 4.0 * sigma xmax = mu + 4.0 * sigma ymax = 1.1 * logistic(mu) #mode of logistic PDF used set key set nozeroaxis set xrange [xmin: xmax] set yrange [0 : ymax] set xlabel "x ->" set ylabel "probability density ->" set xtics xmin, (xmax - xmin)/ 10.0, xmax 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) nohead set arrow from mu, normal(mu + sigma) \ to mu + sigma, normal(mu + sigma) nohead set label "mu" at mu + 1, ymax / 10 set label "sigma" at mu + 1 + sigma, normal(mu + sigma) plot logistic(x), normal(x) pause -1 "Hit return to continue" set noarrow set nolabel # Poisson PDF using normal approximation mu = 5.0 sigma = sqrt(mu) xmin = floor(mu - 4.0 * sigma) xmin = xmin < 0 ? 0 : xmin xmax = ceil(mu + 4.0 * sigma) xinc = ceil((xmax - xmin) / 10) xinc = xinc > 1 ? xinc : 1 ymax = 1.1 * poisson(mu) #mode of poisson PDF used set key set nozeroaxis set xrange [xmin - 1 : xmax + 1] set yrange [0 : ymax] set xlabel "k, x ->" set ylabel "probability density ->" set xtics xmin, xinc, xmax 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 + 0.5, 0 to mu + 0.5, normal(mu) nohead set arrow from mu + 0.5, normal(mu + sigma) \ to mu + 0.5 + sigma, normal(mu + sigma) nohead set label "mu" at mu + 1, ymax / 10 set label "sigma" at mu + 1 + sigma, normal(mu + sigma) plot poisson(x), normal(x - 0.5) set noarrow set nolabel load "defaults.ini"