# (50) Probability distributions¶

This example presents visually the various probability distributions available in gmtmath (as well as grdmath). We evaluate and display both the probability density function (pdf) and the cumulative distribution function (cdf) for each case. The left column shows the density functions while the right column shows the cumulative functions. In addition to the distributions you can also use gmtmath to evaluate critical values for the distributions for chosen confidence levels.

```#!/bin/bash
#               GMT EXAMPLE 50
#
# Purpose:      Illustrate different statistical distributions in gmtmath
# GMT modules:  gmtmath, gmtset, psxy, pstext
#

ps=example_50.ps
# Left column have all the PDFs
gmt set FONT_ANNOT_PRIMARY 10p,Helvetica,black
# Binomial distribution
gmt math -T0/8/1 0.25 8 T BPDF = p.txt
gmt psxy -R-0.6/8.6/0/0.35 -JX3i/0.5i -P -K -Glightgreen p.txt -Sb0.8u -W0.5p -BWS -Bxa1 -Byaf -X1i -Y0.8i > \$ps
# Poisson distribution
gmt math -T0/8/1 T 2 PPDF = p.txt
gmt psxy -R-0.6/8.6/0/0.3 -J -O -K -Glightgreen p.txt -Sb0.8u -W0.5p -BWS -Bxa1 -Byaf -Y0.9i >> \$ps
# Plot normal distribution
gmt math -T-4/4/0.1 T ZPDF = p.txt
gmt psxy -R-4/4/0/0.4 -J -O -K p.txt -L+yb -Glightgreen -W1p -BWS -Bxa1 -Byaf -Y0.9i >> \$ps
# Plot t distribution
gmt psxy -R-4/4/0/0.4 -J -O -K p.txt -W1p,lightgray -BWS -Bxa1 -Byaf -Y0.9i >> \$ps
gmt math -T-4/4/0.1 T 3 TPDF = p.txt
gmt psxy -R -J -O -K p.txt -L+yb -Glightgreen -W1p >> \$ps
# Plot F distribution
gmt math -T0/6/0.02 T 20 12 FPDF = p.txt
gmt psxy -R0/6/0/1 -J -O -K p.txt -L+yb -Glightgreen -W1p -BWS -Bxa1 -Byaf -Y0.9i >> \$ps
# Plot Laplace distribution
gmt math -T-4/4/0.1 T LPDF = p.txt
gmt psxy -R-4/4/0/0.5 -J -O -K p.txt -L+yb -Glightgreen -W1p -BWS -Bxa1 -Byaf -Y0.9i >> \$ps
# Plot Exponential distribution
gmt math -T0/4/0.1 T 2 EPDF = p.txt
gmt psxy -R0/4/0/2.0 -J -O -K p.txt -L+yb -Glightgreen -W1p -BWS -Bxa1 -Byaf -Y0.9i >> \$ps
# Plot Rayleigh distribution
gmt math -T0/6/0.1 T RPDF = p.txt
gmt psxy -R0/6/0/0.7 -J -O -K p.txt -L+yb -Glightgreen -W1p -BWS -Bxa1 -Byaf -Y0.9i >> \$ps
# Plot Weibull distribution
gmt math -T0/6/0.1 T 1 1.5 WPDF = p.txt
gmt psxy -R0/6/0/0.8 -J -O -K p.txt -L+yb -Glightgreen -W1p -BWS -Bxa1 -Byaf -Y0.9i >> \$ps
# Plot Chi-squared distribution
gmt math -T0/12/0.1 T 4 CHI2PDF = p.txt
gmt psxy -R0/12/0/0.20 -J -O -K p.txt -L+yb -Glightgreen -W1p -BWS -Bxa1 -Byaf -Y0.9i >> \$ps
# Right column has all the CDF
# Plot binomial cumulative distribution
gmt math -T0/8/1 0.25 8 T BCDF = p.txt
gmt psxy -R-0.6/8.6/0/1 -J -O -K -Glightred p.txt -Sb0.8u -W0.5p -BES -Bxa1 -Byaf -X3.5i -Y-8.1i >> \$ps
# Plot Poisson cumulative distribution
gmt math -T0/8/1 T 2 PCDF = p.txt
gmt psxy -R-0.6/8.6/0/1 -J -O -K -Glightred p.txt -Sb0.8u -W0.5p -BES -Bxa1 -Byaf -Y0.9i >> \$ps
# Plot normal cumulative distribution
gmt math -T-4/4/0.1 T ZCDF = p.txt
gmt psxy -R-4/4/0/1 -J -O -K p.txt -L+yb -Glightred -W1p -BES -Bxa1 -Byaf -Y0.9i >> \$ps
# Plot t cumulative distribution
gmt psxy -R-4/4/0/1 -J -O -K p.txt -W1p,lightgray -BES -Bxa1 -Byaf -Y0.9i >> \$ps
gmt math -T-4/4/0.1 T 3 TCDF = p.txt
gmt psxy -R -J -O -K p.txt -L+yb -Glightred -W1p >> \$ps
# Plot F cumulative distribution
gmt math -T0/6/0.02 T 20 12 FCDF = p.txt
gmt psxy -R0/6/0/1 -J -O -K p.txt -L+yb -Glightred -W1p -BES -Bxa1 -Byaf -Y0.9i >> \$ps
# Plot Laplace cumulative distribution
gmt math -T-4/4/0.1 T LCDF = p.txt
gmt psxy -R-4/4/0/1  -J -O -K p.txt -L+yb -Glightred -W1p -BES -Bxa1 -Byaf -Y0.9i >> \$ps
# Plot Exponential cumulative distribution
gmt math -T0/4/0.1 T 2 ECDF = p.txt
gmt psxy -R0/4/0/1 -J -O -K p.txt -L+yb -Glightred -W1p -BES -Bxa1 -Byaf -Y0.9i >> \$ps
# Plot Rayleigh cumulative distribution
gmt math -T0/6/0.1 T RCDF = p.txt
gmt psxy -R0/6/0/1 -J -O -K p.txt -L+yb -Glightred -W1p -BES -Bxa1 -Byaf -Y0.9i >> \$ps
# Plot Weibull cumulative distribution
gmt math -T0/6/0.1 T 1 1.5 WCDF = p.txt
gmt psxy -R0/6/0/1 -J -O -K p.txt -L+yb -Glightred -W1p -BES -Bxa1 -Byaf -Y0.9i >> \$ps
# Plot Chi-squared cumulative distribution
gmt math -T0/12/0.1 T 4 CHI2CDF = p.txt
gmt psxy -R0/12/0/1 -J -O -K p.txt -L+yb -Glightred -W1p -BES -Bxa1 -Byaf -Y0.9i >> \$ps
echo "Probability @;lightgreen;Density@;; and @;lightred;Cumulative@;; Distribution Functions" | gmt pstext -R0/6.5/0/1.25 -Jx1i -N -X-3.5i -O -K -F+f18p+cTC >> \$ps
gmt pstext -R0/6.5/0/10 -J -O -F+f14p,Times-Italic+jTC -Dj0.35i -N -Y-8.1i << EOF >> \$ps
3.25 0.9 Binomial P@-8,0.25@-
3.25 1.8 Poisson P(@~l=2@~)
3.25 2.7 Normal P(z)
3.25 3.6 Student t(@~n=3@~)
3.25 4.5 F(@~n@-1@-=20, n@-2@- = 12@~)
3.25 5.4 Laplace P(z)
3.25 6.3 Exponential P(@~l=2@~)
3.25 7.2 Rayleigh P(z)
3.25 8.1 Weibull P(z,1,1.5)
3.25 9 @~c@~@+2@+(z,@~n=4@~)
EOF
rm -f p.txt
```

Probability distributions.