Gnuplot code:

1. Density functions for exponential order statistics (n=5)

set terminal svg size 1300 975 fname "DejaVu Sans" fsize 20 set output "OrderStatistics_Exponential_PDF.svg"

set termopt enhanced

unset key

set xlabel "x" font "DejaVu Sans, 24" set ylabel "f_{X_{(i)}}(x)" font "DejaVu Sans, 24"

1. beta density functions:

b15 (t) = 5 * (1-t)**4 b24 (t) = 20 * t * (1-t)**3 b33 (t) = 30 * (t * (1-t))**2 b42 (t) = 20 * t**3 * (1-t) b51 (t) = 5 * t**4

1. exponential scale parameter (characteristic life) for standard form:

eta = 1.0

1. exponential pdf and cdf:

pdf (t) = eta * exp (-eta * t) cdf (t) = 1 - exp (-eta * t)

set xrange [0:5] set yrange [0:2.3] set format y "%0.1f"

set label 1 "i = 1" at 0.25,2.10 set label 2 "i = 2" at 0.48,1.30 set label 3 "i = 3" at 0.83,0.92 set label 4 "i = 4" at 1.25,0.68 set label 5 "i = 5" at 2.40,0.37

set style line 1 lt 1 lc rgb "#0000FF" lw 3.6 set style line 2 lt 1 lc rgb "#00FF00" lw 3.6 set style line 3 lt 1 lc rgb "#FF0000" lw 3.6 set style line 4 lt 1 lc rgb "#000000" lw 3.6 set style line 5 lt 1 lc rgb "#AA00FF" lw 3.6

plot b15 (cdf (x)) * pdf (x) with lines ls 1 title "i = 1", \

    b24 (cdf (x)) * pdf (x) with lines ls 2 title "i = 2", \
    b33 (cdf (x)) * pdf (x) with lines ls 3 title "i = 3", \
    b42 (cdf (x)) * pdf (x) with lines ls 4 title "i = 4", \
    b51 (cdf (x)) * pdf (x) with lines ls 5 title "i = 5"

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