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GNUPLOT-3.7 demo: prob.dem

#
# $Id: prob.dem,v 1.4 1996/12/09 15:58:09 drd Exp $
#
# Demo Statistical Functions version 2.3
#
# Permission granted to distribute freely for non-commercial purposes only
#
# Copyright (c) 1991, 1992 Jos van der Woude, jvdwoude@hut.nl

print "                   Statistical Library Demo, version 2.3"
print "\n          Copyright (c) 1991, 1992, Jos van de Woude, jvdwoude@hut.nl"
print "Permission granted to distribute freely for non-commercial purposes only"
print "\n\n\n\n\n\n\n"
print "NOTE: contains 54 plots and consequently takes a lot of time to run"
print "                      Press Ctrl-C to exit right now"
pause -1 "                      Press Return to start demo ..."

load "stat.inc"

# Arcsinus PDF and CDF

r = 2.0
mu = 0.0
sigma = r / sqrt2
xmin = -r
xmax = r
ymax = 1.1 * r #No mode
set nokey
set zeroaxis
set xrange [xmin : xmax]
set yrange [0 : ymax]
set xlabel "x ->"
set ylabel "probability density ->"
set xtics autofreq
set ytics autofreq
set format x "%.1f"
set format y "%.1f"
set sample 200
set title "arcsin PDF with r = 2.0"
plot arcsin(x)

all-dem105

pause -1 "Hit return to continue"

set title "arcsin CDF with r = 2.0"
set yrange [0 : 1.1]
plot carcsin(x)

all-dem106

pause -1 "Hit return to continue"

# Beta PDF and CDF
#p = 0.5; q = 0.7
#mu = p / (p + q)
#sigma = sqrt(p**q) / ((p + q ) * sqrt(p + q + 1.0))
#xmin = 0.0
#xmax = 1.0
#Mode of beta PDF used
#ymax = (p < 1.0 || q < 1.0) ? 2.0 : 1.1 * beta((p - 1.0)/(p + q - 2.0))
set key
set zeroaxis
#set xrange [xmin : xmax]
#set yrange [0 : ymax]

set xlabel "x ->"
set ylabel "probability density ->"
set xtics autofreq
set ytics autofreq
set format x "%.1f"
set format y "%.1f"
set sample 100
set title "beta PDF"
plot [0:1] [0:5] p = 0.5, q = 0.7, beta(x) title "p = 0.5, q = 0.7", \
                 p = 5.0, q = 3.0, beta(x) title "p = 5.0, q = 3.0", \
                 p = 0.5, q = 2.5, beta(x) title "p = 0.5, q = 2.5"

all-dem107

pause -1 "Hit return to continue"

set title "incomplete beta CDF"
plot [0:1] [0:1.1] p = 0.5, q = 0.7, cbeta(x) title "p = 0.5, q = 0.7", \
                   p = 5.0, q = 3.0, cbeta(x) title "p = 5.0, q = 3.0", \
                   p = 0.5, q = 2.5, cbeta(x) title "p = 0.5, q = 2.5"

all-dem108

pause -1 "Hit return to continue"

# Binomial PDF and CDF

n = 25; p = 0.15
mu = n * p
sigma = sqrt(n * p * (1.0 - p))
xmin = int(mu - 4.0 * sigma)
xmin = xmin < 0 ? 0 : xmin
xmax = int(mu + 4.0 * sigma)
ymax = 1.1 * binom(mu) #Mode of normal PDF used
xinc = ceil((xmax - xmin) / 10)
xinc = xinc > 1 ? xinc : 1
set nokey
set nozeroaxis
set xrange [xmin : xmax]
set yrange [0 : ymax]
set xlabel "k ->"
set ylabel "probability density ->"
set xtics xmin + 0.499, xinc, xmax
set ytics 0, ymax / 10, ymax
set format x "%2.0f"
set format y "%3.2f"
set sample (xmax - xmin) + 1
set title "binomial PDF with n = 25, p = 0.15"
plot binom(x) with steps

all-dem109

pause -1 "Hit return to continue"

set title "binomial CDF with n = 25, p = 0.15"
set yrange [0 : 1.1]
set ytics 0, 1.1 / 10.5, 1.1
plot cbinom(x) with steps

all-dem110

pause -1 "Hit return to continue"

# Cauchy PDF and CDF
#a = 0.0; b = 2.0
#cauchy PDF has no moments
#xmin = a - 4.0 * b
#xmax = a + 4.0 * b
#ymax = 1.1 * cauchy(a) #Mode of cauchy PDF used
set key
set zeroaxis
#set xrange [xmin : xmax]
#set yrange [0 : ymax]

set xlabel "x ->"
set ylabel "probability density ->"
set xtics autofreq
set ytics autofreq
set format x "%.1f"
set format y "%.2f"
set sample 100
set title "cauchy PDF"
plot [-15:15] [0:0.2] a = 0, b = 2, cauchy(x) title "a = 0, b = 2", \
                      a = 0, b = 4, cauchy(x) title "a = 0, b = 4"

all-dem111

pause -1 "Hit return to continue"

set title "cauchy CDF"
plot [-30:30] [0:1.1] a = 0, b = 2, ccauchy(x) title "a = 0, b = 2", \
                      a = 0, b = 4, ccauchy(x) title "a = 0, b = 4"

all-dem112

pause -1 "Hit return to continue"

# Chi-square PDF and CDF
#df1 = 4.0
#mu = df1
#sigma = sqrt(2.0 * df1)
#xmin = mu - 4.0 * sigma
#xmin = xmin < 0 ? 0 : xmin
#xmax = mu + 4.0 * sigma
#ymax = 1.1 * (df1 > 2.0 ? chi(df1 - 2.0) : 1.0) #Mode of chi PDF used
set key
set zeroaxis
#set xrange [xmin : xmax]
#set yrange [0 : ymax]

set xlabel "x ->"
set ylabel "probability density ->"
set xtics autofreq
set ytics autofreq
set format x "%.1f"
set format y "%.2f"
set sample 100
set title "chi-square PDF"
plot [0:15] [0:0.2] df1 = 4, chi(x) title "df = 4", \
                    df1 = 6, chi(x) title "df = 6", \
                    df1 = 8, chi(x) title "df = 8"

all-dem113

pause -1 "Hit return to continue"

set title "chi-square CDF"
plot [0:15] [0:1.1] df1 = 4, cchi(x) title "df = 4", \
                    df1 = 6, cchi(x) title "df = 6", \
                    df1 = 8, cchi(x) title "df = 8"

all-dem114

pause -1 "Hit return to continue"

# Erlang PDF and CDF
#lambda = 1.0; n = 2.0
#mu = n / lambda
#sigma = sqrt(n) / lambda
#xmin = mu - 4.0 * sigma
#xmin = xmin < 0 ? 0 : xmin
#xmax = mu + 4.0 * sigma
#ymax = n < 2.0 ? 1.0 : 1.1 * erlang((n - 1.0) / lambda) #Mode of erlang PDF used
set key
set zeroaxis
#set xrange [xmin : xmax]
#set yrange [0 : ymax]

set xlabel "x ->"
set ylabel "probability density ->"
set xtics autofreq
set ytics autofreq
set format x "%.1f"
set format y "%.1f"
set sample 100
set title "erlang PDF"
plot [0:10] [0:1] lambda = 1, n = 2, erlang(x) title "lambda = 1, n = 2", \
                  lambda = 2, n = 2, erlang(x) title "lambda = 2, n = 2"

all-dem115

pause -1 "Hit return to continue"

set title "erlang CDF"
plot [0:10] [0:1.1] lambda = 1, n = 2, cerlang(x) title "lambda = 1, n = 2", \
                    lambda = 2, n = 2, cerlang(x) title "lambda = 2, n = 2"

all-dem116

pause -1 "Hit return to continue"

# Thanks to mrb2j@kelvin.seas.Virginia.EDU for telling us about this.
# Extreme (Gumbel extreme value) PDF and CDF
#alpha = 0.5; u = 1.0
#mu = u + (0.577215665/alpha)   # Euler's constant
#sigma = pi/(sqrt(6.0)*alpha)
#xmin = mu - 4.0 * sigma
#xmax = mu + 4.0 * sigma
#ymax = 1.1 * extreme(u) #Mode of extreme PDF used
set key
set zeroaxis
#set xrange [xmin : xmax]
#set yrange [0 : ymax]

set xlabel "x ->"
set ylabel "probability density ->"
set xtics autofreq
set ytics autofreq
set format x "%.1f"
set format y "%.2f"
set sample 100
set title "extreme PDF"
plot [-10:10] [0:0.4] alpha = 0.5, u = 1.0, extreme(x) title "alpha = 0.5, u = 1.0", \
                      alpha = 1.0, u = 0.0, extreme(x) title "alpha = 1.0, u = 0.0"

all-dem117

pause -1 "Hit return to continue"

set title "extreme CDF"
plot [-10:10] [0:1.1] alpha = 0.5, u = 1.0, cextreme(x) title "alpha = 0.5, u = 1.0", \
                      alpha = 1.0, u = 0.0, cextreme(x) title "alpha = 1.0, u = 0.0"

all-dem118

pause -1 "Hit return to continue"

# F PDF and CDF
#df1 = 5.0; df2 = 9.0
#mu = df2 < 2.0 ? 1.0 : df2 / (df2 - 2.0)
#sigma = df2 < 4.0 ? 1.0 : mu * sqrt(2.0 * (df1 + df2 - 2.0) / (df1 * (df2 - 4.0)))
#xmin = mu - 4.0 * sigma
#xmin = xmin < 0 ? 0 : xmin
#xmax = mu + 4.0 * sigma
#Mode of F PDF used
#ymax = df1 < 3.0 ? 1.0 : 1.1 * f((df1 / 2.0 - 1.0) / (df1 / 2.0 + df1 / df2))
set key
set zeroaxis
#set xrange [xmin : xmax]
#set yrange [0 : ymax]

set xlabel "x ->"
set ylabel "probability density ->"
set xtics autofreq
set ytics autofreq
set format x "%.1f"
set format y "%.2f"
set sample 100
set title "F PDF"
plot [0:4] [0:0.8] df1 = 5.0, df2 = 9.0, f(x) title "df1 = 5, df2 = 9", \
                   df1 = 7.0, df2 = 6.0, f(x) title "df1 = 7, df2 = 6"

all-dem119

pause -1 "Hit return to continue"

set title "F CDF"
plot [0:4] [0:1.1] df1 = 5.0, df2 = 9.0, cf(x) title "df1 = 5, df2 = 9", \
                   df1 = 7.0, df2 = 6.0, cf(x) title "df1 = 7, df2 = 6"

all-dem120

pause -1 "Hit return to continue"

# Gamma PDF and incomplete gamma CDF
#rho = 0.5; lambda = 1.0
#mu = rho / lambda
#sigma = sqrt(rho) / lambda
#xmin = mu - 4.0 * sigma
#xmin = xmin < 0 ? 0 : xmin
#xmax = mu + 4.0 * sigma
#ymax = rho < 1.0 ? 2.0 : 1.1 * g((rho - 1.0) / lambda) #Mode of gamma pdf used
set key
set zeroaxis
#set xrange [xmin: xmax]
#set yrange [0: ymax]

set xlabel "x ->"
set ylabel "probability density ->"
set xtics autofreq
set ytics autofreq
set format x "%.1f"
set format y "%.1f"
set sample 100
set title "gamma PDF"
plot [0:5] [0:1.5] rho = 0.5, lambda = 1.0, g(x) title "rho = 0.5, lambda = 1.0", \
                   rho = 1.0, lambda = 1.0, g(x) title "rho = 1.0, lambda = 1.0", \
                   rho = 2.0, lambda = 2.0, g(x) title "rho = 2.0, lambda = 2.0"

all-dem121

pause -1 "Hit return to continue"

set title "incomplete gamma CDF (lambda == 1.0)"
plot [0:5] [0:1.1] rho = 0.5, cgamma(x) title "rho = 0.5", \
                   rho = 1.0, cgamma(x) title "rho = 1.0", \
                   rho = 2.0, cgamma(x) title "rho = 2.0"

all-dem122

pause -1 "Hit return to continue"

# Geometric PDF and CDF

p = 0.4
mu = (1.0 - p) / p
sigma = sqrt(mu / 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 * geometric(mu - 1/p) #mode of gamma PDF used
set nokey
set nozeroaxis
set xrange [xmin : xmax]
set yrange [0 : ymax]
set xlabel "k ->"
set ylabel "probability density ->"
set xtics xmin + 0.499, xinc, xmax
set ytics 0, ymax / 10, ymax
set format x "%2.0f"
set format y "%3.2f"
set sample (xmax - xmin) + 1
set title "geometric PDF with p = 0.4"
plot geometric(x) with steps

all-dem123

pause -1 "Hit return to continue"

set title "geometric CDF with p = 0.4"
set yrange [0 : 1.1]
set ytics 0, 1.1 / 10.5, 1.1
plot cgeometric(x) with steps

all-dem124

pause -1 "Hit return to continue"

# Half normal PDF and CDF

mu = sqrt2invpi
sigma = 1.0
s = sigma*sqrt(1.0 - 2.0/pi)
xmin = 0.0
xmax = mu + 4.0 * s
ymax = 1.1 * halfnormal(0) #Mode of half normal PDF used
set nokey
set zeroaxis
set xrange [xmin: xmax]
set yrange [0: ymax]
set xlabel "x ->"
set ylabel "probability density ->"
set xtics autofreq
set ytics autofreq
set format x "%.1f"
set format y "%.1f"
set sample 100
set title "half normal PDF, sigma = 1.0"
plot halfnormal(x)

all-dem125

pause -1 "Hit return to continue"

set title "half normal CDF, sigma = 1.0"
set yrange [0:1.1]
plot chalfnormal(x)

all-dem126

pause -1 "Hit return to continue"

# Hypergeometric PDF and CPF

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 = 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 * hypgeo(mu) #mode of binomial PDF used
set nokey
set nozeroaxis
set xrange [xmin : xmax]
set yrange [0 : ymax]
set xlabel "k ->"
set ylabel "probability density ->"
set xtics xmin + 0.499, xinc, xmax
set ytics 0, ymax / 10, ymax
set format x "%2.0f"
set format y "%3.2f"
set sample (xmax - xmin) + 1
set title "hypergeometric PDF with nn = 75, mm = 25, n = 10"
plot hypgeo(x) with steps

all-dem127

pause -1 "Hit return to continue"

set yrange [0 : 1.1]
set ytics 0, 1.1 / 10.5, 1.1
set title "hypergeometric CDF with nn = 75, mm = 25, n = 10"
plot chypgeo(x) with steps

all-dem128

pause -1 "Hit return to continue"

# Laplace PDF

a = 0.0; b = 1.0
mu = a
sigma = sqrt(2.0) * b
xmin = mu - 4.0 * sigma
xmax = mu + 4.0 * sigma
ymax = 1.1 * laplace(a) #Mode of laplace PDF used
set nokey
set zeroaxis
set xrange [xmin: xmax]
set yrange [0: ymax]
set xlabel "x ->"
set ylabel "probability density ->"
set xtics autofreq
set ytics autofreq
set format x "%.1f"
set format y "%.2f"
set sample 100
set title "laplace (or double exponential) PDF with a = 0, b = 1"
plot laplace(x)

all-dem129

pause -1 "Hit return to continue"

set title "laplace (or double exponential) CDF with a = 0, b = 1"
set yrange [0: 1.1]
plot claplace(x)

all-dem130

pause -1 "Hit return to continue"

# Logistic PDF and CDF

a = 0.0; lambda = 2.0
mu = a
sigma = pi / (sqrt(3.0) * lambda)
xmin = mu - 4.0 * sigma
xmax = mu + 4.0 * sigma
ymax = 1.1 * logistic(mu) #Mode of logistic PDF used
set nokey
set zeroaxis
set xrange [xmin: xmax]
set yrange [0: ymax]
set nokey
set zeroaxis
set xlabel "x ->"
set ylabel "probability density ->"
set xtics autofreq
set ytics autofreq
set format x "%.1f"
set format y "%.1f"
set sample 100
set title "logistic PDF with a = 0, lambda = 2"
plot logistic(x)

all-dem131

pause -1 "Hit return to continue"

set title "logistic CDF with a = 0, lambda = 2"
set yrange [0: 1.1]
plot clogistic(x)

all-dem132

pause -1 "Hit return to continue"

# Lognormal PDF and CDF

mu = 1.0; sigma = 0.5
m = exp(mu + 0.5 * sigma**2)
s = sqrt(exp(2.0 * mu + sigma**2) * (2.0 * exp(sigma) - 1.0))
xmin = m - 4.0 * s
xmin = xmin < 0 ? 0 : xmin
xmax = m + 4.0 * s
ymax = 1.1 * lognormal(exp(mu - sigma**2)) #Mode of lognormal PDF used
set nokey
set zeroaxis
set xrange [xmin: xmax]
set yrange [0: ymax]
set xlabel "x ->"
set ylabel "probability density ->"
set xtics autofreq
set ytics autofreq
set format x "%.2f"
set format y "%.2f"
set sample 100
set title "lognormal PDF with mu = 1.0, sigma = 0.5"
plot lognormal(x)

all-dem133

pause -1 "Hit return to continue"

set title "lognormal CDF with mu = 1.0, sigma = 0.5"
set yrange [0: 1.1]
plot clognormal(x)

all-dem134

pause -1 "Hit return to continue"

# Maxwell PDF
#a = 0.1
#mu = 2.0 / sqrt(pi) / a
#sigma = sqrt(3.0 - 8.0/pi) / a
#xmin = mu - 4.0 * sigma
#xmin = xmin < 0 ? 0 : xmin
#xmax = mu + 4.0 * sigma
#ymax = 1.1 * maxwell(1.0 / a) #Mode of maxwell PDF used
set key
set zeroaxis
#set xrange[xmin: xmax]
#set yrange[0: ymax]

set xlabel "x ->"
set ylabel "probability density ->"
set xtics autofreq
set ytics autofreq
set format x "%.1f"
set format y "%.1f"
set sample 100
set title "maxwell PDF"
plot [0:6] [0:1.4] a = 1.5, maxwell(x) title "a = 1.5", \
                   a = 1.0, maxwell(x) title "a = 1.0", \
                   a = 0.5, maxwell(x) title "a = 0.5"

all-dem135

pause -1 "Hit return to continue"

set title "maxwell CDF"
plot [0:6] [0:1.1] a = 1.5, cmaxwell(x) title "a = 1.5", \
                   a = 1.0, cmaxwell(x) title "a = 1.0", \
                   a = 0.5, cmaxwell(x) title "a = 0.5"

all-dem136

pause -1 "Hit return to continue"

# Negative binomial PDF and CDF

r = 8; p = 0.4
mu = r * (1.0 - p) / p
sigma = sqrt(mu / 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 * negbin(mu - 1.0/p) #mode of gamma PDF used
set nokey
set nozeroaxis
set xrange [xmin : xmax]
set yrange [0 : ymax]
set xlabel "k ->"
set ylabel "probability density ->"
set xtics xmin + 0.499, xinc, xmax
set ytics 0, ymax / 10, ymax
set format x "%2.0f"
set format y "%3.2f"
set sample (xmax - xmin) + 1
set title "negative binomial (or pascal or polya) PDF with r = 8, p = 0.4"
plot negbin(x) with steps

all-dem137

pause -1 "Hit return to continue"

set yrange [0 : 1.1]
set ytics 0, 1.1 / 10.5, 1.1
set title "negative binomial (or pascal or polya) CDF with r = 8, p = 0.4"
plot cnegbin(x) with steps

all-dem138

pause -1 "Hit return to continue"

# Negative exponential PDF and CDF

lambda = 2.0
mu = 1.0 / lambda
sigma = 1.0 / lambda
xmax =  mu + 4.0 * sigma
ymax = lambda #No mode
set nokey
set zeroaxis
set xrange [0: xmax]
set yrange [0: ymax]
set xlabel "x ->"
set ylabel "probability density ->"
set xtics autofreq
set ytics autofreq
set format x "%.2f"
set format y "%.1f"
set sample 100
set title "negative exponential (or exponential) PDF with lambda = 2.0"
plot nexp(x)

all-dem139

pause -1 "Hit return to continue"

set title "negative exponential (or exponential) CDF with lambda = 2.0"
set yrange [0: 1.1]
plot cnexp(x)

all-dem140

pause -1 "Hit return to continue"

# Normal PDF and CDF
#mu = 0.0; sigma = 1.0
#xmin = mu - 4.0 * sigma
#xmax = mu + 4.0 * sigma
#ymax = 1.1 * normal(mu) #Mode of normal PDF used
set key
set zeroaxis
#set xrange [xmin: xmax]
#set yrange [0: ymax]

set xlabel "x ->"
set ylabel "probability density ->"
set xtics autofreq
set ytics autofreq
set format x "%.1f"
set format y "%.1f"
set sample 100
set title "normal (also called gauss or bell-curved) PDF"
plot [-4:4] [0:1] mu = 0, sigma = 1.0, normal(x) title "mu = 0, sigma = 1.0", \
                  mu = 2, sigma = 0.5, normal(x) title "mu = 2, sigma = 0.5", \
                  mu = 1, sigma = 2.0, normal(x) title "mu = 1, sigma = 2.0"

all-dem141

pause -1 "Hit return to continue"

set title "normal (also called gauss or bell-curved) CDF"
plot [-4:4] [0:1.1] mu = 0, sigma = 1.0, cnormal(x) title "mu = 0, sigma = 1.0", \
                    mu = 2, sigma = 0.5, cnormal(x) title "mu = 2, sigma = 0.5", \
                    mu = 1, sigma = 2.0, cnormal(x) title "mu = 1, sigma = 2.0"

all-dem142

pause -1 "Hit return to continue"

# Pareto PDF and CDF

a = 1.0; b = 3.0
mu = a * b / (b - 1.0)
sigma = a * sqrt(b) / (sqrt(b - 2.0) * (b - 1.0))
xmin = mu - 4.0 * sigma
xmin = xmin < 0 ? 0 : xmin
xmax = mu + 4.0 * sigma
ymax = 1.1 * pareto(a) #mode of pareto PDF used
set nokey
set zeroaxis
set xrange [xmin: xmax]
set yrange [0: ymax]
set xlabel "x ->"
set ylabel "probability density ->"
set xtics autofreq
set ytics autofreq
set format x "%.1f"
set format y "%.1f"
set sample 500
set title "pareto PDF with a = 1, b = 3"
plot pareto(x)

all-dem143

pause -1 "Hit return to continue"

set title "pareto CDF with a = 1, b = 3"
set yrange [0: 1.1]
plot cpareto(x)

all-dem144

pause -1 "Hit return to continue"

# Poisson PDF and CDF

mu = 4.0
sigma = sqrt(mu)
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 * poisson(mu) #mode of poisson PDF used
set nokey
set nozeroaxis
set xrange [xmin : xmax]
set yrange [0 : ymax]
set xlabel "k ->"
set ylabel "probability density ->"
set xtics xmin + 0.499, xinc, xmax
set ytics 0, ymax / 10, ymax
set format x "%2.0f"
set format y "%3.2f"
set sample (xmax - xmin) + 1
set title "poisson PDF with mu = 4.0"
plot poisson(x) with steps

all-dem145

pause -1 "Hit return to continue"

set yrange [0 : 1.1]
set ytics 0, 1.1 / 10.5, 1.1
set title "poisson CDF with mu = 4.0"
plot cpoisson(x) with steps

all-dem146

pause -1 "Hit return to continue"

# Rayleigh PDF and CDF

lambda = 2.0
mu = 0.5 * sqrt(pi / lambda)
sigma = sqrt((1.0 - pi / 4.0) / lambda)
xmax = mu + 4.0 * sigma
ymax = 1.1 * rayleigh(1.0 / sqrt(2.0 * lambda)) #Mode of rayleigh PDF used
set nokey
set zeroaxis
set xrange [0: xmax]
set yrange [0: ymax]
set xlabel "x ->"
set ylabel "probability density ->"
set xtics autofreq
set ytics autofreq
set format x "%.2f"
set format y "%.1f"
set sample 100
set title "rayleigh PDF with lambda = 2.0"
plot rayleigh(x)

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set title "rayleigh CDF with lambda = 2.0"
set yrange [0: 1.1]
plot crayleigh(x)

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# Sine PDF and CDF
#a = 3.0; n = 2
#mu = a / 2.0
#sigma = sqrt(a * a / 3.0 * (1.0 - 3.0 / (2.0 * n * n * pi * pi)) - mu * mu)
#xmin = 0.0
#xmax = a
#ymax = 1.1 * 2.0 / a #Mode of sine PDF used
set key
set zeroaxis
#set xrange [xmin: xmax]
#set yrange [0: ymax]

set xlabel "x ->"
set ylabel "probability density ->"
set xtics autofreq
set ytics autofreq
set format x "%.2f"
set format y "%.1f"
set sample 100
set title "sine PDF"
plot [0:2] [0:1.1] a = 2.0, n = 1, sine(x) title "a = 2.0, n = 1", \
                   a = 2.0, n = 3, sine(x) title "a = 2.0, n = 3"

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set title "sine CDF"
plot [0:2] [0:1.1] a = 2.0, n = 1, csine(x) title "a = 2.0, n = 1", \
                   a = 2.0, n = 3, csine(x) title "a = 2.0, n = 3"

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# t PDF and CDF

df1 = 3.0
mu = 0.0
sigma = df1 > 2.0 ? sqrt(df1 / (df1 - 2.0)) : 1.0
xmin = mu - 4.0 * sigma
xmax = mu + 4.0 * sigma
ymax = 1.1 * t(mu) #Mode of t PDF used
set nokey
set zeroaxis
set xrange [xmin: xmax]
set yrange [0: ymax]
set xlabel "x ->"
set ylabel "probability density ->"
set xtics autofreq
set ytics autofreq
set format x "%.1f"
set format y "%.2f"
set sample 100
set title "t PDF with df1 = 3.0"
plot t(x)

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set title "t CDF with df1 = 3.0"
set yrange [0: 1.1]
plot ct(x)

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# Thanks to efrank@upenn5.hep.upenn.edu for telling us about this
# triangular PDF and CDF

m = 3.0
g = 2.0
mu = m
sigma = g/sqrt(6.0)
xmin = m - g
xmax = m + g
ymax = 1.1 * triangular(m) #Mode of triangular PDF used
set nokey
set zeroaxis
set xrange [xmin: xmax]
set yrange [0: ymax]
set xlabel "x ->"
set ylabel "probability density ->"
set xtics autofreq
set ytics autofreq
set format x "%.1f"
set format y "%.2f"
set sample 100
set title "triangular PDF with m = 3.0, g = 2.0"
plot triangular(x)

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set title "triangular CDF with m = 3.0, g = 2.0"
set yrange [0: 1.1]
plot ctriangular(x)

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# Uniform PDF and CDF

a = -2.0; b= 2.0
mu = (a + b) / 2.0
sigma = (b - a) / sqrt(12.0)
xmin = a
xmax = b
ymax = 1.1 * uniform(mu) #No mode
set nokey
set zeroaxis
set xrange [xmin: xmax]
set yrange [0: ymax]
set xlabel "x ->"
set ylabel "probability density ->"
set xtics autofreq
set ytics autofreq
set format x "%.2f"
set format y "%.2f"
set sample 100
set title "uniform PDF with a = -2.0, b = 2.0"
plot uniform(x)

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set title "uniform CDF with a = -2.0, b = 2.0"
set yrange [0: 1.1]
plot cuniform(x)

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# Weibull PDF and CDF
#lambda = 1.0; n = 1.5
#mu = lambda**(-1.0 / n) * gamma(1.0 / n) / n
#sigma = sqrt(2.0 * lambda**(-2.0 / n) * gamma(2.0 / n) / n - mu * mu)
#xmin = mu - 4.0 * sigma
#xmin = xmin < 0 ? 0 : xmin
#xmax = mu + 4.0 * sigma
#Mode of weibull PDF used
#ymax = 1.1 * (n > 1.0 ? weibull(((n - 1.0) / (lambda * n))**(1.0 / n)) : 2.0)
set key
set zeroaxis
#set xrange [xmin : xmax]
#set yrange [0: ymax]

set xlabel "x ->"
set ylabel "probability density ->"
set xtics autofreq
set ytics autofreq
set format x "%.2f"
set format y "%.1f"
set sample 100
set title "weibull PDF"
plot [0:2] [0:1.5] lambda = 1, n = 0.5, weibull(x) title "lambda = 1, n = 0.5", \
                   lambda = 1, n = 1.0, weibull(x) title "lambda = 1, n = 1.0", \
                   lambda = 1, n = 2.0, weibull(x) title "lambda = 1, n = 2.0", \
                   lambda = 3, n = 2.0, weibull(x) title "lambda = 3, n = 2.0"

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set title "weibull CDF"
plot [0:3] [0:1.2] lambda = 1, n = 0.5, cweibull(x) title "lambda = 1, n = 0.5", \
                   lambda = 1, n = 1.0, cweibull(x) title "lambda = 1, n = 1.0", \
                   lambda = 1, n = 2.0, cweibull(x) title "lambda = 1, n = 2.0", \
                   lambda = 3, n = 2.0, cweibull(x) title "lambda = 3, n = 2.0"

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