Generando Variables Aleatorias
Juan Diego Fuentes
2025-03-26
Transformada Inversa
El método de transformada inversa es una técnica utilizada para generar valores aleatorios a partir de una distribución de probabilidad dada. Se basa en la idea de transformar una variable aleatoria uniforme 𝑈 ∼ 𝑈 ( 0 , 1 ) U∼U(0,1) en una variable aleatoria 𝑋 X con la distribución deseada.
- Ejemplo de generar variables aleatorias de una exponencial con parametros \(\beta = 3\)
a <- 317
c <- 15
m <- 571
X_n <- 41 # semilla
random.number<-numeric(50) # vector numérico de longitud 50
for (i in 1:100)
{X_n<-(a*X_n+c)%%m
random.number[i]<-X_n/m # números en el intervalo [0,1]
}
random.number## [1] 0.78809107 0.85113835 0.83712785 0.39579685 0.49387040 0.58318739
## [7] 0.89667250 0.27145359 0.07705779 0.45359019 0.81436077 0.17863398
## [13] 0.65323993 0.10332750 0.78108581 0.63047285 0.88616462 0.94045534
## [19] 0.15061296 0.77057793 0.29947461 0.95971979 0.25744308 0.63572680
## [25] 0.55166375 0.90367776 0.49211909 0.02802102 0.90893170 0.15761821
## [31] 0.99124343 0.25043783 0.41506130 0.60070053 0.44833625 0.14886165
## [37] 0.21541156 0.31173380 0.84588441 0.17162872 0.43257443 0.15236427
## [43] 0.32574431 0.28721541 0.07355517 0.34325744 0.83887916 0.95096322
## [49] 0.48161121 0.69702277 0.98248687 0.47460595 0.47635727 0.03152364
## [55] 0.01926445 0.13309982 0.21891419 0.42206655 0.82136602 0.39929947
## [61] 0.60420315 0.55866900 0.12434326 0.44308231 0.48336252 0.25218914
## [67] 0.97022767 0.58844133 0.56217163 0.23467601 0.41856392 0.71103327
## [73] 0.42381786 0.37653240 0.38704028 0.71803853 0.64448336 0.32749562
## [79] 0.84238179 0.06129597 0.45709282 0.92469352 0.15411559 0.88091068
## [85] 0.27495622 0.18739054 0.42907180 0.04203152 0.35026270 0.05954466
## [91] 0.90192644 0.93695271 0.04028021 0.79509632 0.07180385 0.78809107
## [97] 0.85113835 0.83712785 0.39579685 0.49387040# generando una variable exponencial con beta = 3
x <- -3*log(1-random.number)
round(x,1) # redondeando a una cifra decimal## [1] 4.7 5.7 5.4 1.5 2.0 2.6 6.8 1.0 0.2 1.8 5.1 0.6 3.2 0.3 4.6
## [16] 3.0 6.5 8.5 0.5 4.4 1.1 9.6 0.9 3.0 2.4 7.0 2.0 0.1 7.2 0.5
## [31] 14.2 0.9 1.6 2.8 1.8 0.5 0.7 1.1 5.6 0.6 1.7 0.5 1.2 1.0 0.2
## [46] 1.3 5.5 9.0 2.0 3.6 12.1 1.9 1.9 0.1 0.1 0.4 0.7 1.6 5.2 1.5
## [61] 2.8 2.5 0.4 1.8 2.0 0.9 10.5 2.7 2.5 0.8 1.6 3.7 1.7 1.4 1.5
## [76] 3.8 3.1 1.2 5.5 0.2 1.8 7.8 0.5 6.4 1.0 0.6 1.7 0.1 1.3 0.2
## [91] 7.0 8.3 0.1 4.8 0.2 4.7 5.7 5.4 1.5 2.0data.frame(1:length(random.number),random.number,round(x,1))## X1.length.random.number. random.number round.x..1.
## 1 1 0.78809107 4.7
## 2 2 0.85113835 5.7
## 3 3 0.83712785 5.4
## 4 4 0.39579685 1.5
## 5 5 0.49387040 2.0
## 6 6 0.58318739 2.6
## 7 7 0.89667250 6.8
## 8 8 0.27145359 1.0
## 9 9 0.07705779 0.2
## 10 10 0.45359019 1.8
## 11 11 0.81436077 5.1
## 12 12 0.17863398 0.6
## 13 13 0.65323993 3.2
## 14 14 0.10332750 0.3
## 15 15 0.78108581 4.6
## 16 16 0.63047285 3.0
## 17 17 0.88616462 6.5
## 18 18 0.94045534 8.5
## 19 19 0.15061296 0.5
## 20 20 0.77057793 4.4
## 21 21 0.29947461 1.1
## 22 22 0.95971979 9.6
## 23 23 0.25744308 0.9
## 24 24 0.63572680 3.0
## 25 25 0.55166375 2.4
## 26 26 0.90367776 7.0
## 27 27 0.49211909 2.0
## 28 28 0.02802102 0.1
## 29 29 0.90893170 7.2
## 30 30 0.15761821 0.5
## 31 31 0.99124343 14.2
## 32 32 0.25043783 0.9
## 33 33 0.41506130 1.6
## 34 34 0.60070053 2.8
## 35 35 0.44833625 1.8
## 36 36 0.14886165 0.5
## 37 37 0.21541156 0.7
## 38 38 0.31173380 1.1
## 39 39 0.84588441 5.6
## 40 40 0.17162872 0.6
## 41 41 0.43257443 1.7
## 42 42 0.15236427 0.5
## 43 43 0.32574431 1.2
## 44 44 0.28721541 1.0
## 45 45 0.07355517 0.2
## 46 46 0.34325744 1.3
## 47 47 0.83887916 5.5
## 48 48 0.95096322 9.0
## 49 49 0.48161121 2.0
## 50 50 0.69702277 3.6
## 51 51 0.98248687 12.1
## 52 52 0.47460595 1.9
## 53 53 0.47635727 1.9
## 54 54 0.03152364 0.1
## 55 55 0.01926445 0.1
## 56 56 0.13309982 0.4
## 57 57 0.21891419 0.7
## 58 58 0.42206655 1.6
## 59 59 0.82136602 5.2
## 60 60 0.39929947 1.5
## 61 61 0.60420315 2.8
## 62 62 0.55866900 2.5
## 63 63 0.12434326 0.4
## 64 64 0.44308231 1.8
## 65 65 0.48336252 2.0
## 66 66 0.25218914 0.9
## 67 67 0.97022767 10.5
## 68 68 0.58844133 2.7
## 69 69 0.56217163 2.5
## 70 70 0.23467601 0.8
## 71 71 0.41856392 1.6
## 72 72 0.71103327 3.7
## 73 73 0.42381786 1.7
## 74 74 0.37653240 1.4
## 75 75 0.38704028 1.5
## 76 76 0.71803853 3.8
## 77 77 0.64448336 3.1
## 78 78 0.32749562 1.2
## 79 79 0.84238179 5.5
## 80 80 0.06129597 0.2
## 81 81 0.45709282 1.8
## 82 82 0.92469352 7.8
## 83 83 0.15411559 0.5
## 84 84 0.88091068 6.4
## 85 85 0.27495622 1.0
## 86 86 0.18739054 0.6
## 87 87 0.42907180 1.7
## 88 88 0.04203152 0.1
## 89 89 0.35026270 1.3
## 90 90 0.05954466 0.2
## 91 91 0.90192644 7.0
## 92 92 0.93695271 8.3
## 93 93 0.04028021 0.1
## 94 94 0.79509632 4.8
## 95 95 0.07180385 0.2
## 96 96 0.78809107 4.7
## 97 97 0.85113835 5.7
## 98 98 0.83712785 5.4
## 99 99 0.39579685 1.5
## 100 100 0.49387040 2.0hist(x, breaks = 20, probability = TRUE, col = "lightblue", ylim=c(0,0.35), main = "Distribución Exponencial (beta = 3)", xlab = "Valor de X")
# Añadimos la función de densidad teórica de la distribución exponencial
curve(dexp(x,rate=1/3), col = "red", lwd = 2, add = TRUE)
Simulando Variables Aleatorias con funciones propias de R
- Generando variable aleatoria uniforme
# Generar 100 variables aleatorias uniformes entre 0 y 1
uniform_random_01 <- runif(100, min = 0, max = 1)
uniform_random_01## [1] 0.549336787 0.678099861 0.530762485 0.418186092 0.347423793 0.954834715
## [7] 0.982722536 0.320687802 0.027474460 0.278048850 0.396454090 0.590104893
## [13] 0.614186120 0.982510939 0.677378194 0.942323952 0.938076384 0.695193199
## [19] 0.572538672 0.004519613 0.687111585 0.104118224 0.024727312 0.384303484
## [25] 0.433817580 0.266134617 0.647565697 0.493335744 0.735969213 0.795003109
## [31] 0.578456266 0.313899508 0.640155830 0.469308808 0.684835513 0.011324642
## [37] 0.489533709 0.904872423 0.895645313 0.344952173 0.381967718 0.500580297
## [43] 0.381902895 0.359902173 0.144014309 0.707601397 0.854677419 0.264958739
## [49] 0.737561780 0.418117836 0.044812228 0.115427511 0.138352373 0.356154714
## [55] 0.879825468 0.220161027 0.751502792 0.097464172 0.602119891 0.430244899
## [61] 0.271144408 0.672071239 0.236696003 0.704617109 0.129269674 0.313063985
## [67] 0.147295280 0.377945638 0.805931550 0.868939239 0.243002580 0.960651330
## [73] 0.074201730 0.457570229 0.575644832 0.537052281 0.823201836 0.530062543
## [79] 0.266102010 0.111477716 0.050086226 0.627102587 0.774124813 0.678765761
## [85] 0.938825893 0.318877882 0.858626687 0.514461691 0.104774268 0.632743567
## [91] 0.981254340 0.677676253 0.471779206 0.478251905 0.589236320 0.768460107
## [97] 0.988432239 0.676723337 0.203754146 0.670547446- Generando una variable aleatoria de una distribucion normal
# Generar 100 variables normales con media 30 y desviación estándar 3.5
normal_random <- rnorm(100, mean = 30, sd = 3.5)
normal_random## [1] 30.09904 26.29316 34.94318 29.40490 29.63721 22.95658 29.00918 35.54842
## [9] 32.70026 35.47148 28.60669 34.95912 31.78040 31.41578 29.34253 28.50933
## [17] 30.16759 29.36341 31.77078 33.81286 27.60367 22.61941 31.66103 27.90272
## [25] 32.31789 34.57128 32.29555 33.19365 28.84368 33.04370 38.31616 30.02133
## [33] 26.29525 25.71510 25.60011 34.94580 29.29027 30.80082 27.02712 30.69506
## [41] 31.93067 26.89217 30.79210 29.73449 30.99980 29.73203 30.46079 30.76027
## [49] 27.54173 37.07919 28.08562 20.19537 29.14635 28.81219 28.33861 30.44170
## [57] 27.17499 34.06354 31.11443 27.48803 29.72605 29.58243 27.76193 33.56888
## [65] 27.46696 32.43936 30.04127 29.22498 33.09665 34.67938 28.19154 19.28708
## [73] 22.28724 30.11496 27.29933 29.54083 26.41998 26.23768 24.44167 28.55004
## [81] 29.58292 29.37674 26.69502 32.46162 33.42406 34.28255 33.57550 26.65568
## [89] 29.89018 34.87903 34.28818 28.95424 29.38893 36.59199 29.24129 29.09014
## [97] 33.03819 24.18316 30.08053 31.81592hist(normal_random, col=3, main="Grafico de La normal", xlim=c(15,40))
- Generando Variable Aleatoria de una exponencial Para generar variables aleatorias con distribución exponencial con parámetro β . Ejemplo:
Simular El tiempo de servicio en una caja de un banco cuyo comportamiento sigue una forma exponencial con media de 3 minutos/cliente
numexp <- rexp(100,1/3)
round(numexp,1)## [1] 2.1 8.8 1.6 0.9 8.1 0.0 0.3 2.3 0.7 0.3 1.2 3.8 8.7 0.3 0.4
## [16] 3.7 0.2 9.5 0.4 4.9 0.3 0.9 6.6 3.3 7.5 1.7 3.8 2.6 28.4 0.7
## [31] 0.3 1.0 8.0 3.3 3.5 2.3 3.3 12.6 0.6 6.9 0.7 6.4 5.7 1.5 0.6
## [46] 0.6 2.0 1.6 2.2 1.5 1.2 0.7 4.1 0.1 1.1 6.8 1.1 2.2 2.2 1.1
## [61] 5.4 4.0 3.3 3.7 0.4 0.7 1.7 3.2 0.9 2.0 3.7 1.1 3.6 3.7 0.3
## [76] 3.2 4.2 2.1 6.3 0.2 6.8 1.0 2.2 1.5 0.5 0.5 2.7 3.6 3.9 0.9
## [91] 1.3 1.6 1.7 2.4 2.4 0.8 0.0 0.0 4.9 3.7- Generando variable aleatoria de una binomial
binom_random <-rbinom(100,12,0.25)
binom_random ## [1] 4 2 2 2 2 4 0 3 5 2 1 6 3 2 1 5 0 4 3 2 0 5 2 5 2 3 5 4 0 3 2 3 4 2 1 3 3
## [38] 2 4 4 1 2 3 3 2 5 5 1 2 1 6 2 3 1 3 1 5 2 2 3 3 4 3 1 6 3 0 1 4 5 1 3 2 3
## [75] 3 3 1 1 2 2 4 1 5 2 6 1 2 4 2 1 1 2 1 3 3 4 2 4 3 3- Generando variables aleatorias de una poisson
evento_pois <- rpois(100,5)
tablapois <- table(evento_pois)
tablapois## evento_pois
## 0 1 2 3 4 5 6 7 8 9 10 11 12 18
## 1 7 7 7 15 20 15 14 5 2 3 2 1 1evento_pois## [1] 2 5 6 3 4 7 11 4 4 3 5 2 10 1 1 2 7 7 4 8 5 1 5 18 4
## [26] 3 4 12 5 6 3 5 7 5 5 8 6 6 2 5 7 7 6 6 9 6 5 4 9 4
## [51] 7 5 7 10 4 3 4 6 5 1 2 6 10 5 5 6 5 5 1 3 5 4 8 4 5
## [76] 3 6 7 7 4 7 6 6 5 1 6 11 8 0 2 2 7 8 6 4 7 1 4 5 7