Distribución normal
Jhonier Rangel
2025-05-27
Teorema del límite central
Promedios o sumas de variables i.i.d. cuando la varianza es finita y acotada tiende a una normal
Problema de Galton
y<-NULL
for(i in 1:1000){
x<-2*rbinom(100,1,0.5)-1
x
y[i]<-sum(x)
}
y## [1] -12 4 4 8 -2 -2 -12 -12 -14 16 -6 10 -6 -6 -26 16 0 -14
## [19] 12 4 22 -8 18 -20 -20 -14 -6 14 10 4 4 -12 2 -6 4 -4
## [37] 2 14 -22 0 -16 -12 2 14 4 6 2 12 -16 -12 14 12 -4 8
## [55] -12 8 6 0 14 -12 -26 18 -10 -8 2 0 8 20 -2 4 -8 0
## [73] 0 12 6 0 -18 -4 14 -10 2 2 8 4 -10 10 -2 -4 20 -2
## [91] -6 -12 0 4 -14 10 2 -2 -8 0 6 8 4 -6 0 -2 24 -12
## [109] 16 22 14 4 -10 2 6 8 -18 -4 -8 4 24 4 -10 -10 -20 0
## [127] 0 -8 -2 2 -8 0 -16 10 8 -2 -4 -6 12 10 -8 -2 2 16
## [145] 6 4 -8 0 -6 12 -4 4 0 4 -8 -10 -6 2 2 -20 12 6
## [163] 4 4 0 -2 14 2 -2 0 -16 8 -2 -14 -2 4 2 -20 8 -8
## [181] -4 -16 -16 10 0 14 10 14 -2 10 -10 4 -6 -18 -2 -6 16 10
## [199] 6 2 -8 6 -8 6 -10 18 -14 -6 -8 4 10 -6 -8 -10 4 -4
## [217] 2 -20 -6 8 -18 20 -18 6 -4 -8 6 8 4 -10 -2 -12 -8 -4
## [235] -10 -4 -6 0 2 16 0 -26 -10 10 -4 -12 4 -10 -22 -2 0 2
## [253] -18 -12 -10 -2 -10 -4 -14 4 -6 -12 8 8 0 10 8 -8 4 -10
## [271] -6 8 -14 -10 -2 4 -24 4 -6 -2 -4 4 -10 -6 18 -22 10 12
## [289] 14 -4 8 -2 8 -6 8 -14 -2 -8 -2 -2 -28 0 18 8 16 10
## [307] -12 -6 6 -8 -22 -14 -12 -16 2 -14 20 0 -10 0 2 6 0 -4
## [325] 0 4 -14 14 12 0 20 28 -4 8 -4 0 -4 12 10 0 4 -16
## [343] -12 2 12 10 -4 22 10 -4 -10 0 -4 -6 10 2 22 -4 36 -16
## [361] 0 4 -6 6 -14 6 -8 -10 2 4 4 -10 -8 -12 8 -12 2 10
## [379] 2 0 6 2 -10 0 4 2 14 -4 8 -4 -8 2 8 2 8 10
## [397] -2 4 2 0 -2 -10 0 12 8 2 -4 -4 2 2 14 0 4 -10
## [415] -20 -12 -14 2 0 -12 0 4 6 0 0 -8 14 2 0 -12 4 -16
## [433] -2 6 8 4 0 -6 -4 -2 22 8 0 -16 8 -8 6 6 -12 4
## [451] -22 4 6 -4 8 2 -12 -12 2 4 -8 0 0 -4 -12 2 -2 10
## [469] 4 -2 6 0 10 -2 0 -6 8 -6 0 2 -4 -14 -10 4 6 8
## [487] 32 6 -6 4 0 2 -8 -6 -6 6 -4 -4 4 -6 -8 -4 0 4
## [505] -4 2 -6 12 12 8 -4 -14 -12 -4 -10 8 2 10 -10 -2 -6 -16
## [523] 2 8 8 -6 10 12 6 16 14 0 -2 0 -14 -14 -8 -14 -12 -2
## [541] 2 -4 -16 20 0 -4 -10 16 12 2 4 -4 12 6 -10 -12 -6 22
## [559] -6 -10 16 -10 -2 14 -6 10 2 -8 4 4 14 0 -4 0 10 -6
## [577] -18 -8 10 -10 -12 2 -4 -8 -8 8 6 8 2 2 -10 -20 14 12
## [595] 8 -2 12 -10 -10 6 2 -8 4 4 8 -24 8 26 -6 -8 -2 -6
## [613] -2 8 -6 -4 6 6 -10 -10 4 -20 -12 6 0 -6 18 -2 18 -14
## [631] 4 2 -6 10 4 -10 -2 16 -14 8 -6 -8 0 8 4 4 0 8
## [649] -12 -24 4 4 -4 10 4 8 10 10 -2 2 6 0 -4 -8 -6 8
## [667] -12 -8 -8 16 12 8 -12 22 -6 6 4 2 -8 4 2 -2 16 -8
## [685] -8 -8 0 -12 -2 -16 -6 4 0 -14 -12 -10 4 -8 -6 2 2 -4
## [703] 4 -12 -2 -18 -2 2 -20 0 -4 0 2 -10 8 -2 -20 12 -14 -4
## [721] -4 14 0 -4 0 18 12 -8 -16 0 0 -6 -18 -20 2 4 14 20
## [739] 14 10 -2 -4 -12 4 -2 2 0 -4 12 -4 -12 4 -6 -8 -12 -16
## [757] 20 18 8 -2 -12 -12 2 6 -4 2 12 -4 -16 -14 -4 6 -4 4
## [775] -18 8 4 -6 6 -14 16 14 -14 -10 0 -12 4 -4 16 -2 4 4
## [793] 0 -2 -12 -4 6 -22 -10 -4 2 6 -2 2 -6 -4 -2 2 2 4
## [811] -4 -6 -14 -2 2 -4 -4 -12 0 6 -14 10 -12 8 -12 10 8 0
## [829] 14 10 -4 2 8 4 0 -12 4 10 4 -2 -14 -6 8 -4 -2 4
## [847] -4 0 -6 10 -6 -4 -6 -22 4 4 -8 4 -18 -16 2 -2 -6 -8
## [865] -16 0 0 -6 -10 4 4 -12 14 -2 -4 8 6 6 2 -12 -6 -4
## [883] 16 -4 0 2 -10 4 4 -4 12 0 8 2 -8 4 6 -6 6 -2
## [901] 18 8 -6 2 -18 -2 -10 0 16 2 -6 10 4 10 6 -14 12 22
## [919] 2 -4 6 8 10 0 -18 4 -12 0 -12 4 4 0 2 -16 -16 6
## [937] -4 14 4 -2 6 -4 -20 2 0 -6 10 12 12 10 -18 -8 -22 10
## [955] -8 2 -12 2 -8 -10 -8 10 0 10 -4 -6 10 -4 12 14 -8 0
## [973] 10 10 -24 -2 -6 4 12 24 0 -14 8 16 12 -8 -2 -16 -12 6
## [991] -20 -4 -2 4 -2 -6 -16 -8 14 -4hist(y)
plot(density(y))
Ejercicios
r: Random
p: Porbablity
d: Density
x<-seq(-5,5,length=1000)
d1<-dnorm(x)
d2<-dnorm(x,0,1.5)
d3<-dnorm(x,0,2)
plot(x,d1,col="blue")+lines(x,d2,col="red")+lines(x,d3,col="green")
## integer(0)- Dada una variable aleatoria continua Z, con distribución normal estándar, es decir, N(0;1), encuentre las siguientes probabilidades, usando la tabla. a) (0 ≤ ≤1,25). Rpta: 0,3944 b) ( ≥1,25). Rpta: 0,1056 c) ( ≤−1,25). Rpta: 0,1056 d) (0≤ ≤1,33). Rpta: 0,4082 e) ( ≥1,33). Rpta: 0,0918 f) (−1,33 ≤ ≤0). Rpta: 0,4082
pnorm(1.25,0,1)-pnorm(0,0,1)## [1] 0.39435021-pnorm(1.25,0,1)## [1] 0.1056498pnorm(-1.25,0,1)## [1] 0.1056498pnorm(1.33,0,1)-pnorm(0,0,1)## [1] 0.4082409- El peso de cierto modelo de baterías está normalmente distribuido con una media de 6g y desviación estándar de 2g. Determine el porcentaje de baterías cuyo peso es mayor de 8g.
1-pnorm(8,6,2)## [1] 0.1586553- Los precios de las acciones de cierta industria se distribuyen en forma normal con media de $20 y desviación estándar de $3. ¿Cuál es la probabilidad de que el precio de las acciones de una empresa se encuentre entre $18 y $20?
pnorm(20,20,3)-pnorm(18,20,3)## [1] 0.2475075x<-seq(12,28,length=1000)
d<-dnorm(x,20,3)
plot(x,d,type="l",col="#a613d9")+lines(abline(v=20), col = "#a613d9")+lines(abline(v=18), col = "#a613d9")
## integer(0)Cuantiles: q
Si X es una variable aleatoria continua, distribuida de forma normal, con media de 18 y varianza de 6,25. Encontrar: a) el valor de a, tal que ( ≥ ) =0,1814. Rpta: 20,275 b) el valor de c, tal que ( < )=0,2236.
qnorm(1-0.1814,18,sqrt(6.25))## [1] 20.27511qnorm(0.2236,18,sqrt(6.25))## [1] 16.09977