############################################################# lambda <- 0.5 y.grid.1 <- seq( 0, 10, length = 500 ) plot( y.grid.1, dexp( y.grid.1, lambda ), type = 'l', xlab = 'y', ylab = 'Density', lwd = 2, main = 'Exponential PDF (lambda = 0.5)' ) ############################################################# F.Y <- function( y, lambda ) { CDF <- ifelse( y < 0, 0, 1 - exp( - lambda * y ) ) return( CDF ) } y.grid.2 <- seq( -5, 15, length = 500 ) plot( y.grid.2, F.Y( y.grid.2, lambda ), type = 'l', xlab = 'y', ylab = 'CDF', lwd = 2, col = 'red', main = 'Exponential CDF (lambda = 0.5)' ) abline( h = 0, lwd = 2 ) abline( h = 1, lwd = 2 ) ############################################################# F.inverse.Y <- function( p, lambda ) { F.inverse <- - log( 1 - p ) / lambda return( F.inverse ) } p.grid.1 <- seq( 0.001, 0.999, length = 500 ) plot( p.grid.1, F.inverse.Y( p.grid.1, lambda ), type = 'l', xlab = 'p', ylab = 'Inverse CDF', lwd = 2, col = 'red', main = 'Exponential Inverse CDF (lambda = 0.5)' ) abline( v = 1, lwd = 2 ) p.grid.2 <- seq( 0, 1, length = 500 ) plot( p.grid.2, F.inverse.Y( p.grid.2, lambda ), type = 'l', xlab = 'p', ylab = 'Inverse CDF', lwd = 2, col = 'red', main = 'Exponential Inverse CDF (lambda = 0.5)' ) abline( v = 1, lwd = 2 ) p.grid.2 [1] 0.000000000 0.002004008 0.004008016 0.006012024 0.008016032 0.010020040 [7] 0.012024048 0.014028056 0.016032064 0.018036072 0.020040080 0.022044088 [13] 0.024048096 0.026052104 0.028056112 0.030060120 0.032064128 0.034068136 [19] 0.036072144 0.038076152 0.040080160 0.042084168 0.044088176 0.046092184 [25] 0.048096192 0.050100200 0.052104208 0.054108216 0.056112224 0.058116232 [31] 0.060120240 0.062124248 0.064128257 0.066132265 0.068136273 0.070140281 [37] 0.072144289 0.074148297 0.076152305 0.078156313 0.080160321 0.082164329 [43] 0.084168337 0.086172345 0.088176353 0.090180361 0.092184369 0.094188377 [49] 0.096192385 0.098196393 0.100200401 0.102204409 0.104208417 0.106212425 [55] 0.108216433 0.110220441 0.112224449 0.114228457 0.116232465 0.118236473 [61] 0.120240481 0.122244489 0.124248497 0.126252505 0.128256513 0.130260521 [67] 0.132264529 0.134268537 0.136272545 0.138276553 0.140280561 0.142284569 [73] 0.144288577 0.146292585 0.148296593 0.150300601 0.152304609 0.154308617 [79] 0.156312625 0.158316633 0.160320641 0.162324649 0.164328657 0.166332665 [85] 0.168336673 0.170340681 0.172344689 0.174348697 0.176352705 0.178356713 [91] 0.180360721 0.182364729 0.184368737 0.186372745 0.188376754 0.190380762 [97] 0.192384770 0.194388778 0.196392786 0.198396794 0.200400802 0.202404810 [103] 0.204408818 0.206412826 0.208416834 0.210420842 0.212424850 0.214428858 [109] 0.216432866 0.218436874 0.220440882 0.222444890 0.224448898 0.226452906 [115] 0.228456914 0.230460922 0.232464930 0.234468938 0.236472946 0.238476954 [121] 0.240480962 0.242484970 0.244488978 0.246492986 0.248496994 0.250501002 [127] 0.252505010 0.254509018 0.256513026 0.258517034 0.260521042 0.262525050 [133] 0.264529058 0.266533066 0.268537074 0.270541082 0.272545090 0.274549098 [139] 0.276553106 0.278557114 0.280561122 0.282565130 0.284569138 0.286573146 [145] 0.288577154 0.290581162 0.292585170 0.294589178 0.296593186 0.298597194 [151] 0.300601202 0.302605210 0.304609218 0.306613226 0.308617234 0.310621242 [157] 0.312625251 0.314629259 0.316633267 0.318637275 0.320641283 0.322645291 [163] 0.324649299 0.326653307 0.328657315 0.330661323 0.332665331 0.334669339 [169] 0.336673347 0.338677355 0.340681363 0.342685371 0.344689379 0.346693387 [175] 0.348697395 0.350701403 0.352705411 0.354709419 0.356713427 0.358717435 [181] 0.360721443 0.362725451 0.364729459 0.366733467 0.368737475 0.370741483 [187] 0.372745491 0.374749499 0.376753507 0.378757515 0.380761523 0.382765531 [193] 0.384769539 0.386773547 0.388777555 0.390781563 0.392785571 0.394789579 [199] 0.396793587 0.398797595 0.400801603 0.402805611 0.404809619 0.406813627 [205] 0.408817635 0.410821643 0.412825651 0.414829659 0.416833667 0.418837675 [211] 0.420841683 0.422845691 0.424849699 0.426853707 0.428857715 0.430861723 [217] 0.432865731 0.434869739 0.436873747 0.438877756 0.440881764 0.442885772 [223] 0.444889780 0.446893788 0.448897796 0.450901804 0.452905812 0.454909820 [229] 0.456913828 0.458917836 0.460921844 0.462925852 0.464929860 0.466933868 [235] 0.468937876 0.470941884 0.472945892 0.474949900 0.476953908 0.478957916 [241] 0.480961924 0.482965932 0.484969940 0.486973948 0.488977956 0.490981964 [247] 0.492985972 0.494989980 0.496993988 0.498997996 0.501002004 0.503006012 [253] 0.505010020 0.507014028 0.509018036 0.511022044 0.513026052 0.515030060 [259] 0.517034068 0.519038076 0.521042084 0.523046092 0.525050100 0.527054108 [265] 0.529058116 0.531062124 0.533066132 0.535070140 0.537074148 0.539078156 [271] 0.541082164 0.543086172 0.545090180 0.547094188 0.549098196 0.551102204 [277] 0.553106212 0.555110220 0.557114228 0.559118236 0.561122244 0.563126253 [283] 0.565130261 0.567134269 0.569138277 0.571142285 0.573146293 0.575150301 [289] 0.577154309 0.579158317 0.581162325 0.583166333 0.585170341 0.587174349 [295] 0.589178357 0.591182365 0.593186373 0.595190381 0.597194389 0.599198397 [301] 0.601202405 0.603206413 0.605210421 0.607214429 0.609218437 0.611222445 [307] 0.613226453 0.615230461 0.617234469 0.619238477 0.621242485 0.623246493 [313] 0.625250501 0.627254509 0.629258517 0.631262525 0.633266533 0.635270541 [319] 0.637274549 0.639278557 0.641282565 0.643286573 0.645290581 0.647294589 [325] 0.649298597 0.651302605 0.653306613 0.655310621 0.657314629 0.659318637 [331] 0.661322645 0.663326653 0.665330661 0.667334669 0.669338677 0.671342685 [337] 0.673346693 0.675350701 0.677354709 0.679358717 0.681362725 0.683366733 [343] 0.685370741 0.687374749 0.689378758 0.691382766 0.693386774 0.695390782 [349] 0.697394790 0.699398798 0.701402806 0.703406814 0.705410822 0.707414830 [355] 0.709418838 0.711422846 0.713426854 0.715430862 0.717434870 0.719438878 [361] 0.721442886 0.723446894 0.725450902 0.727454910 0.729458918 0.731462926 [367] 0.733466934 0.735470942 0.737474950 0.739478958 0.741482966 0.743486974 [373] 0.745490982 0.747494990 0.749498998 0.751503006 0.753507014 0.755511022 [379] 0.757515030 0.759519038 0.761523046 0.763527054 0.765531062 0.767535070 [385] 0.769539078 0.771543086 0.773547094 0.775551102 0.777555110 0.779559118 [391] 0.781563126 0.783567134 0.785571142 0.787575150 0.789579158 0.791583166 [397] 0.793587174 0.795591182 0.797595190 0.799599198 0.801603206 0.803607214 [403] 0.805611222 0.807615230 0.809619238 0.811623246 0.813627255 0.815631263 [409] 0.817635271 0.819639279 0.821643287 0.823647295 0.825651303 0.827655311 [415] 0.829659319 0.831663327 0.833667335 0.835671343 0.837675351 0.839679359 [421] 0.841683367 0.843687375 0.845691383 0.847695391 0.849699399 0.851703407 [427] 0.853707415 0.855711423 0.857715431 0.859719439 0.861723447 0.863727455 [433] 0.865731463 0.867735471 0.869739479 0.871743487 0.873747495 0.875751503 [439] 0.877755511 0.879759519 0.881763527 0.883767535 0.885771543 0.887775551 [445] 0.889779559 0.891783567 0.893787575 0.895791583 0.897795591 0.899799599 [451] 0.901803607 0.903807615 0.905811623 0.907815631 0.909819639 0.911823647 [457] 0.913827655 0.915831663 0.917835671 0.919839679 0.921843687 0.923847695 [463] 0.925851703 0.927855711 0.929859719 0.931863727 0.933867735 0.935871743 [469] 0.937875752 0.939879760 0.941883768 0.943887776 0.945891784 0.947895792 [475] 0.949899800 0.951903808 0.953907816 0.955911824 0.957915832 0.959919840 [481] 0.961923848 0.963927856 0.965931864 0.967935872 0.969939880 0.971943888 [487] 0.973947896 0.975951904 0.977955912 0.979959920 0.981963928 0.983967936 [493] 0.985971944 0.987975952 0.989979960 0.991983968 0.993987976 0.995991984 [499] 0.997995992 1.000000000 F.inverse.Y( 1, lambda ) [1] Inf ############################################################# par( mfrow = c( 2, 1 ) ) n <- 100 seed <- 1 set.seed( seed ) u.star <- runif( n ) y.star <- - log( u.star ) / lambda hist( y.star, breaks = 20, prob = T, xlab = 'y', main = 'Random Draws From Exponential PDF (lambda = 0.5)', xlim = c( 0, 13 ) ) y.grid.3 <- seq( 0, 10, length = 500 ) lines( y.grid.3, dexp( y.grid.3, lambda ), lty = 1, col = 'red', lwd = 2 ) text( 5, 0.3, 'n = 100', cex = 1.1 ) ############################################################# n <- 1000 seed <- 1 set.seed( seed ) u.star <- runif( n ) y.star <- - log( u.star ) / lambda hist( y.star, breaks = 20, prob = T, xlab = 'y', main = 'Random Draws From Exponential PDF (lambda = 0.5)', xlim = c( 0, 13 ) ) y.grid.4 <- seq( 0, 13, length = 500 ) lines( y.grid.4, dexp( y.grid.4, lambda ), lty = 1, col = 'red', lwd = 2 ) text( 5, 0.3, 'n = 1000', cex = 1.1 ) ############################################################# par( mfrow = c( 1, 1 ) ) hist( y.star, breaks = 100, prob = T, xlab = 'y', main = 'Random Draws From Exponential PDF (lambda = 0.5)' ) hist( y.star, prob = T, xlab = 'y', main = 'Random Draws From Exponential PDF (lambda = 0.5)' ) y.grid.4 <- seq( 0, 14, length = 500 ) lines( y.grid.4, dexp( y.grid.4, lambda ), lty = 1, col = 'red', lwd = 2 ) text( 7, 0.3, 'n = 1000', cex = 1.1 ) ############################################################# y.grid.5 <- seq( 0, 50, length = 500 ) lambda <- 0.1 plot( y.grid.5, dexp( y.grid.5, lambda ), type = 'l', xlab = 'y', ylab = 'Density', lwd = 2, main = 'Exponential PDF', col = 'red', ylim = c( 0., 1.0 ) ) abline( h = 0, lwd = 2 ) abline( v = 0, lwd = 2 ) text( 30, 0.075, 'lambda = 0.1', col = 'red', cex = 1.1 ) lambda <- 0.5 lines( y.grid.5, dexp( y.grid.5, lambda ), lwd = 2, main = 'Exponential PDF', col = 'blue' ) text( 10, 0.2, 'lambda = 0.5', col = 'blue', cex = 1.1 ) lambda <- 1.0 lines( y.grid.5, dexp( y.grid.5, lambda ), lwd = 2, main = 'Exponential PDF', col = 'green' ) text( 7.5, 0.8, 'lambda = 1.0', col = 'green', cex = 1.1 ) ############################################################# y.grid.cauchy <- seq( -15, 15, length = 500 ) plot( y.grid.cauchy, dcauchy( y.grid.cauchy ), type = 'l', xlab = 'y', ylab = 'Density', lwd = 2, main = 'Cauchy PDF' ) lines( y.grid.cauchy, dnorm( y.grid.cauchy, 0, 1 / qnorm( 0.75 ) ), lty = 2, lwd = 2, col = 'red' ) #############################################################