Gráfico de las distribuciones de Probabilidad en R

##Para detalles de las distribuciones puede obtener el libro “Modelos de probabilidad” en la siguiente dirección

https://1drv.ms/b/s!Aj-hHTVbsx01hs9cx6tbr9v1vZKjzQ?e=qyn6U4

1. Distribuciones discretas

1.1 Distribución uniforme discreta

\(f(x)=\dfrac{1}{N} \hspace{2cm} x=1,2,3,\cdots N\)

\(E(X)=\dfrac{N+1}{2}\)

\(V(X)= \dfrac{N^2-1}{12}\)

para N=6

x<-c(seq(1,6,1))
p1<-c(rep(1/6,6))
p2<-cumsum(p1)
p3<-c(0,p2)
cbind(x,p1,p2)

##      x        p1        p2
## [1,] 1 0.1666667 0.1666667
## [2,] 2 0.1666667 0.3333333
## [3,] 3 0.1666667 0.5000000
## [4,] 4 0.1666667 0.6666667
## [5,] 5 0.1666667 0.8333333
## [6,] 6 0.1666667 1.0000000

plot(x,p1,type="h",main="Dist. uniforme discreta",lwd=2,ylim=c(0,0.18),ylab="f(x)")

f1<-stepfun(x,p3)
plot(f1,verticals=F,main="Dist. uniforme discreta acumulada",lwd=2,pch=16,ylab="F(x")

1.2 Binomial

\(f(x)=\dbinom{n}{x}p^{x}q^{n-x} \hspace{2cm} x=0,1,2,3,\cdots n\)

\(E(X)=np\)

\(V(X)=npq\)

Para n=10, p=0.2

x<-0:10
y<-dbinom(x,10,0.2)
y1<-pbinom(x,10,0.2)
z<-cbind(x,y,y1)
z

##        x            y        y1
##  [1,]  0 0.1073741824 0.1073742
##  [2,]  1 0.2684354560 0.3758096
##  [3,]  2 0.3019898880 0.6777995
##  [4,]  3 0.2013265920 0.8791261
##  [5,]  4 0.0880803840 0.9672065
##  [6,]  5 0.0264241152 0.9936306
##  [7,]  6 0.0055050240 0.9991356
##  [8,]  7 0.0007864320 0.9999221
##  [9,]  8 0.0000737280 0.9999958
## [10,]  9 0.0000040960 0.9999999
## [11,] 10 0.0000001024 1.0000000

plot(x,y,type="h",xaxt="n",lwd=2,main="Dist. binomial")
axis(1,c(0:10))

plot(stepfun(0:10,pbinom(c(0,0:10),10,0.2)),verticals=F,pch=16,
main="Dist. Binomial acumulada",xaxt="n",ylab="F(x)")
axis(1,c(0:10))

1.3 Distribución hipergeométrica

\(f(x)=\dfrac{\dbinom{k}{x}\dbinom{N-k}{n-x}}{\dbinom{N}{n}}\hspace{2cm} x=0,1,2,3,\cdots n\)

\(E(X)= \dfrac{nk}{N}\)

\(V(X)= \dfrac{nk(N-k)(N-n)}{N^{2}(N-1)}\)

Para N=20, K=5, n=4

x1<-0:4
y2<-dhyper(x1,20,5,4)
y3<-phyper(x1,20,5,4)
z1<-cbind(x1,y2,y3)
z1

##      x1           y2           y3
## [1,]  0 0.0003952569 0.0003952569
## [2,]  1 0.0158102767 0.0162055336
## [3,]  2 0.1501976285 0.1664031621
## [4,]  3 0.4505928854 0.6169960474
## [5,]  4 0.3830039526 1.0000000000

plot(x1,y2,type="h",xaxt="n",lwd=2,main="Dist. Hipergeométrica")
axis(1,c(0:4))

plot(stepfun(0:4,phyper(c(0,0:4),20,5,4)),verticals=F,pch=16,
main="Dist. hiperergeométrica  acumulada",xaxt="n",ylab="F(x)")
axis(1,c(0:10))

1.4 Distribución geométrica

\(f(x)=pq^{x} \hspace{2cm} x=0,1,2,3,\ldots\)

\(E(X)= \dfrac{q}{p}\)

\(V(X)= \dfrac{q}{p^{2}}\)

Para p=0.3

x3<-0:20
y6<-dgeom(x3,0.3)
y7<-pgeom(x3,0.3)
z3<-cbind(x3,y6,y7)
z3

##       x3           y6        y7
##  [1,]  0 0.3000000000 0.3000000
##  [2,]  1 0.2100000000 0.5100000
##  [3,]  2 0.1470000000 0.6570000
##  [4,]  3 0.1029000000 0.7599000
##  [5,]  4 0.0720300000 0.8319300
##  [6,]  5 0.0504210000 0.8823510
##  [7,]  6 0.0352947000 0.9176457
##  [8,]  7 0.0247062900 0.9423520
##  [9,]  8 0.0172944030 0.9596464
## [10,]  9 0.0121060821 0.9717525
## [11,] 10 0.0084742575 0.9802267
## [12,] 11 0.0059319802 0.9861587
## [13,] 12 0.0041523862 0.9903111
## [14,] 13 0.0029066703 0.9932178
## [15,] 14 0.0020346692 0.9952524
## [16,] 15 0.0014242685 0.9966767
## [17,] 16 0.0009969879 0.9976737
## [18,] 17 0.0006978915 0.9983716
## [19,] 18 0.0004885241 0.9988601
## [20,] 19 0.0003419669 0.9992021
## [21,] 20 0.0002393768 0.9994415

plot(x3,y6,type="h",xaxt="n",lwd=2,main="Distribución geométrica")
axis(1,c(0:20))

plot(stepfun(0:20,pgeom(c(0,0:20),0.3)),verticals=F,pch=16,
main="Dist. geométrica acumulada",xaxt="n",ylab="F(x)")
axis(1,c(0:20))

1.5 Distribución binomial negativa

\(f(x)= \dbinom{x+k-1}{k-1}p^{k}q^{x} \hspace{2cm}x=0,1,2,\ldots\)

\(E(X)= \dfrac{kq}{p}\)

\(V(X)=\dfrac{kq}{p^{2}}\)

Para K=3 , p=0.6

x4<-0:15
y8<-dnbinom(x4,3,0.6)
y9<-pnbinom(x4,3,0.6)
z4<-cbind(x4,y8,y9)
z4

##       x4           y8        y9
##  [1,]  0 2.160000e-01 0.2160000
##  [2,]  1 2.592000e-01 0.4752000
##  [3,]  2 2.073600e-01 0.6825600
##  [4,]  3 1.382400e-01 0.8208000
##  [5,]  4 8.294400e-02 0.9037440
##  [6,]  5 4.644864e-02 0.9501926
##  [7,]  6 2.477261e-02 0.9749652
##  [8,]  7 1.274020e-02 0.9877054
##  [9,]  8 6.370099e-03 0.9940755
## [10,]  9 3.114271e-03 0.9971898
## [11,] 10 1.494850e-03 0.9986847
## [12,] 11 7.066563e-04 0.9993913
## [13,] 12 3.297730e-04 0.9997211
## [14,] 13 1.522029e-04 0.9998733
## [15,] 14 6.957847e-05 0.9999429
## [16,] 15 3.154224e-05 0.9999744

plot(x4,y8,type="h",xaxt="n",lwd=2,main="Distribución binomial negativa")
axis(1,c(0:15))

plot(stepfun(0:15,pnbinom(c(0,0:15),3,0.6)),verticals=F,pch=16,
main="Dist. binomial negativa acumulada",xaxt="n",ylab="F(x)")
axis(1,c(0:15))

1.6 Distribución logaritmica

\(f(x)= \dfrac{-1}{ln(1-p)}\cdot\dfrac{p^{x}}{x} \hspace{2cm}x=1,2,3, \ldots\)

\(E(X)=\dfrac{kp}{1-p}\)

\(V(X)=\dfrac{kp(1-kp)}{(1-p)^{2}}\)

siendo \(k=\dfrac{-1}{ln(1-p)}\)

x<-seq(1,10,1)
p<-0.6
f7<--1/(log(1-p))*p^x/x
p7<-cumsum(f7)
cbind(x,f7,p7)

##        x           f7        p7
##  [1,]  1 0.6548140008 0.6548140
##  [2,]  2 0.1964442002 0.8512582
##  [3,]  3 0.0785776801 0.9298359
##  [4,]  4 0.0353599560 0.9651958
##  [5,]  5 0.0169727789 0.9821686
##  [6,]  6 0.0084863894 0.9906550
##  [7,]  7 0.0043644289 0.9950194
##  [8,]  8 0.0022913252 0.9973108
##  [9,]  9 0.0012220401 0.9985328
## [10,] 10 0.0006599016 0.9991927

p8<-c(0,p7)
plot(x,f7,type="h",lwd=2,main="Dist. logaritmica ",ylab="f(x)")

f8<-stepfun(x,p8)
plot(f8,verticals=F,lwd=2,main="Dist. logaritmica acumulada",pch=16,ylab="F(x)")

2. Distribuciones continuas

2.1 Distribución uniforme continua

\(f(x)= \dfrac{1}{b-a}\hspace{2cm} a \leq x \leq b\)

\(E(X)=\dfrac{a+b}{2}\)

\(V(X)=\dfrac{(b-a)^{2}}{12}\)

para a=2 b=5

curve(dunif(x,min=2,max=5),from=1, to=6,lwd=2,main="Distribución uniforme continua",ylab="f(x)")

curve(punif(x,min=2,max=5),from=1,to=6,lwd=2,main="Distribución uniforme continua acumulada",ylab="F(x)")

2.2 Distribución normal

\(f(x)=\dfrac{1}{\sigma \sqrt{2\pi}}e^{-\dfrac{1}{2}\left( \dfrac{ x-\mu}{\sigma}\right) ^{2}} \hspace{1cm} -\infty <x< \infty\)

\(E(X)=\mu\)

\(V(X)=\sigma^2\)

Para \(\mu=50\) \(\sigma= 5\)

curve(dnorm(x,50,5),xlim=c(30,80),lwd=2,ylab="f(x)",main="Dist. normal")

curve(pnorm(x,50,5),xlim=c(30,80),lwd=2,ylab="F(x)",main="Dist. normal acumulada")

2.3 Distribución exponencial

\(f(x)=\dfrac{1}{\theta}exp \left( -\dfrac{x}{\theta}\right) \hspace{1cm} x \geq 0\)

\(E(X)=\theta\)

\(V(X)= \theta^{2}\)

curve(dexp(x,1/5),xlim=c(0,20),lwd=2,ylab="f(x)",main="Distribución exponencial")

curve(pexp(x,1/5),xlim=c(0,20),lwd=2,ylab="F(x)",main="Distribución exponencial acumulada")

2.4 Distribución Gamma

\(f(x)= \dfrac{1}{\Gamma(\alpha)\beta^{\alpha}}x^{\alpha-1}e^{-\dfrac{x}{\beta}} \hspace{1cm} x \geq 0\)

\(E(X)=)=\alpha \beta\)

\(V(X)= \alpha \beta^{2}\)

curve(dgamma(x,4,1/3),xlim=c(0,50),lwd=2,ylab="f(x)",main="Distribución gamma")

curve(pgamma(x,4,1/3),xlim=c(0,50),lwd=2,ylab="F(x)",main="Distribución gamma acumulada")

2.5 Distribución lognormal \(f(x)=\dfrac{1}{x \sigma\sqrt{2\pi}}e^{-\dfrac{1}{2}\left( \dfrac{Ln x-\mu}{\sigma}\right) ^{2}} \hspace{1cm} x \geq 0\)

\(E(X)=e^{\mu+\frac{\sigma^{2}}{2}}\)

\(V(X)=(e^{\sigma^{2}}-1)e^{2\mu+\sigma^{2}}\)

curve(dlnorm(x,2,1/2),xlim=c(0,25),lwd=2,ylab="f(x)",main="Distribución lognormal")

curve(plnorm(x,2,1/2),xlim=c(0,25),lwd=2,ylab="F(x)",main="Distribución lognormal acumulada")

2.6 Distribución beta

\(f(x)=\dfrac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}x^{\alpha-1}(1-x)^{\beta-1}\hspace{1cm} 0 < x <1\)

\(E(X)=\dfrac{\alpha}{\alpha+\beta}\)

\(V(X)=\dfrac{\alpha \beta}{(\alpha+\beta)^{2}(\alpha+\beta+1)}\)

curve(dbeta(x,4,2),xlim=c(0,1),lwd=2,ylab="f(x)",main="Distribución beta")

curve(pbeta(x,4,2),xlim=c(0,1),lwd=2,ylab="F(x)",main="Distribución beta acumulada")

2.7 Distribución Pareto \(f(x)= \dfrac{\alpha}{x}\left( \dfrac{\beta}{x}\right) ^{\alpha} \hspace{1cm} x \geq \beta\)

\(E(X)=\dfrac{\alpha \beta}{\alpha-1}\)

\(V(X)=\dfrac{\alpha \beta^{2}}{(\alpha-1)^{2}(\alpha-2)}\)

library(EnvStats)

## 
## Attaching package: 'EnvStats'

## The following objects are masked from 'package:stats':
## 
##     predict, predict.lm

## The following object is masked from 'package:base':
## 
##     print.default

curve(dpareto(x,100,6.5),xlim=c(90,200),lwd=2,ylab="f(x)",main="Distribución Pareto")

curve(ppareto(x,100,6.5),xlim=c(90,200),lwd=2,ylab="F(x)",main="Distribución Pareto acumulada")

2.8 Distribución triangular

$f(x)= \[\begin{cases} \dfrac{2(x-a)}{(b-a)(c-a)}&\text{si $a \leq x <c$}\\ \dfrac{2(b-x)}{(b-a)(b-c)}&\text{si $ c\leq x \leq b$}\\$0$ &\text{en otro caso} \end{cases}\]

$

\(E(X)=\dfrac{a+b+c}{3}\)

\(V(X)=\dfrac{a^{2}+b^{2}+c^{2}+ab+ac+bc}{6}\)

library(triangle)
curve(dtriangle(x,3,11,8),xlim=c(3,11),lwd=2,ylab="f(x)",main="Distribución triangular")

curve(ptriangle(x,3,11,8),xlim=c(3,11),lwd=2,ylab="F(x)",main="Distribución triangular acumulada")

2.9 Distribución logística

\(f(x)=\dfrac{exp\left[- \left( \dfrac{x-\alpha}{\beta}\right) \right] }{\beta\left\lbrace1+exp\left[ -\left(\dfrac{x-\alpha}{\beta} \right) \right] \right\rbrace ^{2}}\)

\(E(X)= \alpha\)

\(V(X)= \dfrac{\beta^{2} \pi^{2}}{3}\)

curve(dlogis(x,2000,300),xlim=c(0,5000),lwd=2,ylab="f(x)",main="Distribución logística")

curve(plogis(x,2000,300),xlim=c(0,5000),lwd=2,ylab="F(x)",main="Distribución logística acumulada")

2.10 Distribución Pearson III

\(f(x)= \dfrac{1}{ \alpha \Gamma (\beta)}\left( \dfrac{x-\lambda}{\alpha}\right)^{\beta-1}exp\left( - \left( \dfrac{x-\lambda}{\alpha}\right) \right)\)

\(E(X)=\lambda+\alpha \beta\)

\(V(X)=\beta \alpha^{2}\)

library(PearsonDS)
curve(dpearsonIII(x,3,1,0.5),xlim=c(0,10),lwd=2,ylab="f(x)",main="Distribución PearsonIII")

curve(ppearsonIII(x,3,1,0.5),xlim=c(0,10),lwd=2,ylab="F(x)",main="Distribución PearsonIII acumulada")

3. Distribuciones de valore extremos

3.1 Distribución general de valores extremos

\(f(x)=\begin{cases} \dfrac{1}{\sigma}\left[ 1 + \xi \left( \dfrac{x- \mu}{\sigma}\right) \right] ^{-\frac{1}{\xi}- 1} exp \left\lbrace -\left[ 1+\xi \left( \dfrac{x - \mu}{\sigma}\right) \right] ^{-\frac{1}{\xi}}\right\rbrace &\text{ si $\xi \neq 0$}\\ \dfrac{1}{\sigma}exp\left[ - \left( \dfrac{x - \mu}{\sigma}\right) \right] exp\left[ - exp\left[ -\left( \dfrac{x- \mu}{\sigma}\right) \right] \right] &\text{ si $\xi = 0$}\\ \end{cases}\)

\(E(X)= \begin{cases} \mu + \dfrac{\sigma \cdot \Gamma (1- \xi) - 1}{\xi}&\text{ si $\xi \neq 0, \xi < 0$}\\ \mu + 0.5772 \cdot \sigma &\text{ si $\xi = 0$}\\ No \,\,\,existe &\text{ si $\xi \geq 1$} \end{cases}\)

\(V(X)=\begin{cases} \dfrac{\sigma^{2}(g_{2} - g_{1}^{2})}{\xi^{2}}&\text{ si $\xi \neq 0, \xi < \frac{1}{2}$}\\ \dfrac{\sigma ^{2}\pi^{2}}{6} &\text{ si $\xi = 0$}\\ No \,\,existe &\text{ si $\xi \geq \frac{1}{2}$} \end{cases}\)

library(evd)
curve(dgev(x,2,1,0.5),xlim=c(0,10),lwd=2,ylab="f(x)",main="Distribución GVE")

curve(pgev(x,11,3.5,-0.4),xlim=c(0,20),lwd=2,ylab="F(x)",main="Distribución GVE acumulada")

3.2 Distribución Weibull

\(f(x)=\dfrac{\alpha}{\beta}\left( \dfrac{x}{\beta} \right)^{\alpha-1}exp \left( -\left( \dfrac{x}{\beta}\right) ^{\alpha} \right) \hspace{1cm} x \geq 0\)

\(E(X)=\beta \Gamma\left(1+\dfrac{1}{\alpha} \right)\)

\(V(X)=\beta^{2}\left[ \Gamma\left(1+\dfrac{2}{\alpha} \right)-\Gamma^{2}\left( 1+\dfrac{1}{\alpha} \right) \right]\)

curve(dweibull(x,2,4),xlim=c(0,15),lwd=2,ylab="f(x)",main="Distribución Weibull")

curve(pweibull(x,2,4),xlim=c(0,15),lwd=2,ylab="F(x)",main="Distribución Weibull  acumulada")

3.3 Distribución Weibull 3 parámetros

\(f(x)=\dfrac{\alpha}{\beta}\left( \dfrac{x-\theta}{\beta} \right)^{\alpha-1} exp\left(-\left( \dfrac{x-\theta}{\beta}\right) ^{\alpha} \right) \hspace{1cm}x \geq \theta\)

\(E(X)=\theta+\beta \Gamma\left(1+\dfrac{1}{\alpha} \right)\)

\(V(X)=\beta^{2} \left[ \Gamma\left( 1+ \dfrac{2}{\alpha}\right) -\Gamma^{2}\left( 1+\dfrac{1}{\alpha}\right) \right]\)

library(FAdist)

## 
## Attaching package: 'FAdist'

## The following objects are masked from 'package:evd':
## 
##     dgev, dgumbel, pgev, pgumbel, qgev, qgumbel, rgev, rgumbel

## The following objects are masked from 'package:EnvStats':
## 
##     dlnorm3, plnorm3, qlnorm3, rlnorm3

curve(dweibull3(x,5,3,2),xlim=c(2,8),lwd=2,ylab="f(x)",main="Distribución Weibull 3 parámetros")

curve(pweibull3(x,5,3,2),xlim=c(2,8),lwd=2,ylab="F(x)",main="Distribución Weibull 3 parámetros  acumulada")

3.4 Distribución Frechet

\(f(x)= \dfrac{\alpha}{\delta}\left( \dfrac{x-\lambda}{\delta} \right)^{-1 - \alpha} \cdot exp\left( -\left( \dfrac{x - \lambda}{\delta}\right)^{- \alpha} \right) \hspace{1cm}x \geq \lambda\)

\(E(X)=\lambda +\delta\Gamma \left( 1- \dfrac{1}{\alpha} \right)\)

\(V(X)=\delta^{2}\left[ \Gamma\left( 1-\dfrac{2}{\alpha} \right) - \Gamma^{2}\left( 1- \dfrac{1}{\alpha}\right) \right]\)

curve(dfrechet(x,3,0.8,1.5),xlim=c(2,10),lwd=2,ylab="f(x)",main="Distribución Frechet")

curve(pfrechet(x,3,0.8,1.5),xlim=c(2,10),lwd=2,ylab="F(x)",main="Distribución Frechet acumulada")

3.5 Distribución Gumbel

\(f(x)=\dfrac{1}{\beta}exp\left[ - \left( \dfrac{x-\mu}{\beta} \right) exp\left( - \left( \dfrac{x-\mu}{\beta} \right) \right)\right]\)

\(E(X)=\mu + 0.5772\beta\)

\(V(X)=\dfrac{\pi^{2}}{6}\beta^{2}\)

curve(dgumbel(x,3,1.5),xlim=c(-5,15),lwd=2,ylab="f(x)",main="Distribución Gumbel")

curve(pgumbel(x,3,1.5),xlim=c(-5,15),lwd=2,ylab="F(x)",main="Distribución Gumbel acumulada")

3.6 Distribución generalizada de Pareto

\(f(x)=\begin{cases} \dfrac{1}{\sigma}\left( 1+\xi \left( \dfrac{x- \mu}{\sigma}\right) \right)^{-\frac{1}{\xi}-1} &\text{si $\xi \neq 0$}\\ \dfrac{1}{\sigma}exp\left(-\left(\dfrac{x-\mu}{\sigma} \right) \right) &\text{si $\xi = 0$} \end{cases}\)

\(E(X)=\mu + \dfrac{\sigma}{\xi}\)

\(V(X)=\dfrac{\sigma^{2}}{(1-\xi)^{2}(1-2\xi)}\)

curve(dgpd(x,2,0.5,0.8),xlim=c(1,8),lwd=2,ylab="f(x)",main="Distribución GP")

curve(pgpd(x,2,0.5,0.8),xlim=c(1,8),lwd=2,ylab="F(x)",main="Distribución GP acumulada")