Distribuição da Normal

Y ~ N(Mu,Sigma)

X ~ N(500, 100)

W ~ N(10, 2)

Normal padrão

Z ~ N(0,1)

library(RcmdrMisc)  
## Carregando pacotes exigidos: car
## Carregando pacotes exigidos: carData
## Carregando pacotes exigidos: sandwich
.x <- seq(-3.291, 3.291, length.out=1000)    
plotDistr(.x, dnorm(.x, mean=0, sd=1), cdf=FALSE, xlab="z", ylab="Densidade",regions=list(c(0, 4)), col=c('#0080C0', '#BEBEBE'), legend=FALSE)

P(z > 0) = 0,5

P(-1 < z <1) = 0,6826

plotDistr(.x, dnorm(.x, mean=0, sd=1), cdf=FALSE, xlab="z", 
          ylab="Densidade",regions=list(c(-1, 1)), 
          col=c('red'), legend=FALSE)

P(-2 < z < 2) = 0,9544

plotDistr(.x, dnorm(.x, mean=0, sd=1), cdf=FALSE, xlab="z", 
          ylab="Densidade",regions=list(c(-2, 2)), 
          col=c('green'), legend=FALSE)

P(-3 < z < 3) = 0,9974

plotDistr(.x, dnorm(.x, mean=0, sd=1), cdf=FALSE, xlab="z", 
          ylab="Densidade",regions=list(c(-3, 3)), 
          col=c('yellow'), legend=FALSE)

#—————————————————————–

P(0 < z < 1) = 0,6826/2 = 0,3413

P(0 < z < 1) = 0,3413

plotDistr(.x, dnorm(.x, mean=0, sd=1), cdf=FALSE, xlab="z", 
            ylab="Densidade",regions=list(c(0, 1)), 
            col=c('skyblue'), legend=FALSE)

#—————————————————————–

P(0 < z < 2) = 0,9544/2 = 0,4772

plotDistr(.x, dnorm(.x, mean=0, sd=1), cdf=FALSE, xlab="z", 
            ylab="Densidade",regions=list(c(0, 2)), 
            col=c('darkblue'), legend=FALSE)

#—————————————————————–

P(-3 < z < 0) = P(0 < z < 3) = 0,9974/2 = 0,4987

plotDistr(.x, dnorm(.x, mean=0, sd=1), cdf=FALSE, xlab="z", 
              ylab="Densidade",regions=list(c(-3, 0),c(0,3)), 
              col=c('purple','yellow'), legend=FALSE)

#—————————————————————–

P(-2 < z < 0) = P(0 < z < 2) = 0,4772

plotDistr(.x, dnorm(.x, mean=0, sd=1), cdf=FALSE, xlab="z", 
          ylab="Densidade",regions=list(c(-2, 0),c(0,2)), 
          col=c('red','blue'), legend=FALSE)

#—————————————————————– #—————————————————————– #—————————————————————–

P(1 < z < 2) = ?

plotDistr(.x, dnorm(.x, mean=0, sd=1), cdf=FALSE, xlab="z", 
            ylab="Densidade",regions=list(c(1,2)), 
            col=c('darkgreen'), legend=FALSE)

plotDistr(.x, dnorm(.x, mean=0, sd=1), cdf=FALSE, xlab="z", 
          ylab="Densidade",regions=list(c(0,1),c(1,2)), 
          col=c('red','darkgreen'), legend=FALSE)

P(1 < z < 2) = P(0 < z < 2) - P(0 < z <1) = 0,4772 - 0,3413 = 0,1359

P(1 < z < 2) = 0,1359

#————————————————————

Área preta = (preta+vermelha) - vermelha

P(2 <z <3) = P(0<z<3) -P(0<z<2) = 0,4987 - 0,4772 = 0,0215

plotDistr(.x, dnorm(.x, mean=0, sd=1), cdf=FALSE, xlab="z", 
            ylab="Densidade",regions=list(c(2,3)), 
            col=c('black'), legend=FALSE)

plotDistr(.x, dnorm(.x, mean=0, sd=1), cdf=FALSE, xlab="z", 
          ylab="Densidade",regions=list(c(0,2),c(2,3)), 
          col=c('red','black'), legend=FALSE)

#———————————————————-

P(-3<z<-2) = P(2<z<3)=0,0215

plotDistr(.x, dnorm(.x, mean=0, sd=1), cdf=FALSE, xlab="z", 
            ylab="Densidade",regions=list(c(-3,-2)), 
            col=c('orange'), legend=FALSE)

#———————————————————-

Área total = area azul escuro + area azul claro

P(-3<z<2) = P(-3 < z < 0) + P(0 <z <2)

P(-3<z<2) = 0,4987 + 0,4772 = 0,9759

plotDistr(.x, dnorm(.x, mean=0, sd=1), cdf=FALSE, xlab="z", 
          ylab="Densidade",regions=list(c(-3,2)), 
          col=c('royalblue'), legend=FALSE)

plotDistr(.x, dnorm(.x, mean=0, sd=1), cdf=FALSE, xlab="z", 
          ylab="Densidade",regions=list(c(-3,0),c(0,2)), 
          col=c('darkblue','skyblue'), legend=FALSE)

#———————————————————-

P(-2 < z < 1) = 0,8185

P(-2 < z < 1) = P(-2 < z < 0) + P(0<z<1) =

= P(0<z<2) + P(0<z<1) =

= 0,4772 + 0,3413 = 0,8185

plotDistr(.x, dnorm(.x, mean=0, sd=1), cdf=FALSE, xlab="z", 
            ylab="Densidade",regions=list(c(-2,0),c(0,1)), 
            col=c('red','blue'), legend=FALSE)

#———————————————————-

P(0 < z <1,34) = 0,40988

plotDistr(.x, dnorm(.x, mean=0, sd=1), cdf=FALSE, xlab="z", 
          ylab="Densidade",regions=list(c(0,1.34)), 
          col=c('darkred'), legend=FALSE)

TABELA DA NORMAL PADRÃO

#———————————————————-

P(0 < z < 2,13) = 0,48341

plotDistr(.x, dnorm(.x, mean=0, sd=1), cdf=FALSE, xlab="z", 
          ylab="Densidade",regions=list(c(0,2.13)), 
          col=c('blue'), legend=FALSE)

#———————————————————-

P(0 < z< 1.12) = 0,36864

#———————————————————-

AREA TOTAL = VERMELHA + AZUL

P(-2,01 <z < 1,17) = P(-2,01 < z < 0) + P(0<z<1,17)

P(-2,01 <z < 1,17) = P(0 < z < 2,01) + P(0<z<1,17)

P(-2,01 <z < 1,17) = 0,47778 + 0,37900

P(-2,01 <z < 1,17) = 0,85678

plotDistr(.x, dnorm(.x, mean=0, sd=1), cdf=FALSE, xlab="z", 
          ylab="Densidade",regions=list(c(-2.01,0),c(0,1.17)), 
          col=c('red','blue'), legend=FALSE)