library(RcmdrMisc)

## Loading required package: car

## Loading required package: carData

## Loading required package: sandwich

.x <- seq(-3.291, 3.291, length.out=1000)    
plotDistr(.x, dnorm(.x, mean=0, sd=1), cdf=FALSE, xlab="x", ylab="Densidade",regions=list(c(-1, 1)), col=c('#0080C0', '#BEBEBE'), legend=FALSE)

#X ~ N(mu, sigma)

#Distribuição Normal Padrão #X ~ N( 0, 1 )

#Pr(-1<X<1) = 0,6826

plotDistr(.x, dnorm(.x, mean=0, sd=1), cdf=FALSE, xlab="x", ylab="Densidade",regions=list(c(-1, 1)), col=c('#0080C0', '#BEBEBE'), legend=FALSE)

#Pr(-2<X<2) = 0,9544

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

#Pr(-3<X<3) = 0,9974

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

#Pr(0<X<1) = 0,3413

plotDistr(.x, dnorm(.x, mean=0, sd=1), cdf=FALSE, xlab="x", 
          ylab="Densidade",regions=list(c(0, 1)), 
          col=c('#0080C0', '#BEBEBE'), legend=FALSE)

#Pr(-2<X<0) = 0,9544/2 = 0,4772

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

#Pr(0<X<3) = 0,9974/2 = 0,4987

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

#Pr(1<X<2) = Pr(0<X<2) - P(0<X<1) = 0,4772 - 0,3413 = 0,1359

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

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

#Pr(2<X<3) = Pr(0<X<3) - Pr(0<X<2) = 0,4987-0,4772 = 0,0215

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

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

# Pr(-3<X<-2) = 0,0215

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

#Pr(-3<X<-1) = P(-3<X<0) - P(-1<X<0) = 0,4987 - 0,3413 = 0.1574

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

#Pr(-3<X<1) = P(-3<X<0) + P(0<X<1) = 0,4987 + 0,3413 = 0,84

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

# Pr(-2<X<0) = P(0<X<2) = 0,4772

#Pr(0 < X <1,34) = 0,40988

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

#Pr( 0< X< 0,87) = 0,30785

plotDistr(.x, dnorm(.x, mean=0, sd=1), cdf=FALSE, xlab="x", 
          ylab="Densidade",regions=list(c(0,0.87)), 
          col=c('#BEBEBE', '#BEBEBE'), legend=FALSE)

pnorm(0.87) - pnorm(0)

## [1] 0.3078498

P(x>3) =

plotDistr(.x, dnorm(.x, mean=0, sd=1), cdf=FALSE, xlab="x", 
          ylab="Densidade",regions=list(c(3,10)), 
          col=c('skyblue', '#BEBEBE'), legend=FALSE)

plotDistr(.x, dnorm(.x, mean=0, sd=1), cdf=FALSE, xlab="x", 
          ylab="Densidade",regions=list(c(-10,3)), 
          col=c('skyblue', '#BEBEBE'), legend=FALSE)

1 - pnorm(3)

## [1] 0.001349898

pnorm(3, lower.tail =F)

## [1] 0.001349898

#P(x>-2) = 0,9772 = 0,5 + P(0<X<2) = 0,5 + 0,4772

plotDistr(.x, dnorm(.x, mean=0, sd=1), cdf=FALSE, xlab="x", 
            ylab="Densidade",regions=list(c(-2,10)), 
            col=c('skyblue', '#BEBEBE'), legend=FALSE)

plotDistr(.x, dnorm(.x, mean=0, sd=1), cdf=FALSE, xlab="x", 
          ylab="Densidade",regions=list(c(-10,-2)), 
          col=c('skyblue', '#BEBEBE'), legend=FALSE)

#Pr(-0,87<X<2,97) = 0,8063

pnorm(2.97)-pnorm(-0.87)

## [1] 0.8063608

#Pr(0,13 <X<0,48) =

pnorm(0.48)-pnorm(0.13)

## [1] 0.1326695

#Pr(-2,77<X<-1,56) =

pnorm(-1.56)-pnorm(-2.77)

## [1] 0.05657713

Pr(X > -1,46) =

pnorm(-1.46,lower.tail = F)

## [1] 0.927855

plotDistr(.x, dnorm(.x, mean=0, sd=1), cdf=FALSE, xlab="x", 
          ylab="Densidade",regions=list(c(-1.46,10)), 
          col=c('skyblue', '#BEBEBE'), legend=FALSE)

.x <- seq(500, 700, length.out=1000)    
plotDistr(.x, dnorm(.x, mean=589, sd=400), cdf=FALSE, xlab="x", 
          ylab="Densidade",regions=list(c(550,600)), 
          col=c('skyblue', '#BEBEBE'), legend=FALSE)

X ~ N(589, 400) Pr(550<X<600) Z = (X - media)/sigma Z ~ N(0,1)

Pr(550<X<600) = Pr(550-589 <X-media<600-589) = Pr(-39<X - media<11) = Pr(-39/400<(X - media)/sigma<11/400) =
Pr(-0,0975< Z<0,0275) = pnorm(0.0275)-pnorm(-0.0975) = 0.04980486 Pr(550<X<600) = 0.04980486

Y ~ N(10, 5) Pr(8<Y<9) = Pr(8-10<Y-media<9-10) = Pr(-2<Y-media<-1) = Pr(-2/5<(Y-media)/sigma<-1/5) = Pr(-0,4<Z<-0,2) pnorm(-0.2)-pnorm(-0.4) = 0.07616203 Pr(8<Y<9) = 0,0761

W ~ N(47, 30) Pr(50<W<55) = Pr(50-47 <W-media<55-47) = Pr(3 <W-media<8) = Pr(3/30 <Z<8/30) = Pr(0,1 <Z<0,2667) =
pnorm(0.2667)-pnorm(0.1) =0.06532209 Pr(50<W<55) = 0,065322

X ~ N(13, 40) Pr(9<X<14) 0.04980036 Pr(9-13<X-media<14-13) Pr(-4<X-media<1) Pr(-4/40<(X-media)/sigma<1/40) Pr(-0,1<Z<0,025)

.x <- seq(-3.291, 3.291, length.out=1000)    
plotDistr(.x, dnorm(.x, mean=0, sd=1), 
          cdf=FALSE, xlab="x", 
          ylab="Densidade",
          regions=list(c(-0.1, 0.025)), col=c('#0080C0', '#BEBEBE'), legend=FALSE)

pnorm(0.025)-pnorm(-0.1)

## [1] 0.04980036

Y ~ N(5000,4000) Pr(4800<Y<4900) = Pr(4800-5000<Y-media<4900-5000) = Pr(-200<Y-media<-100) = Pr(-200/4000<Z<-100/4000) =
Pr(-0,05<Z<-0,025) =

plotDistr(.x, dnorm(.x, mean=0, sd=1), 
            cdf=FALSE, xlab="x", 
            ylab="Densidade",
            regions=list(c(-0.05, -0.025)), col=c('#0080C0', '#BEBEBE'), legend=FALSE)

pnorm(-0.025)-pnorm(-0.05)

## [1] 0.009966288

W ~ N(70, 50) Pr(70<W< 80) Pr(70-70<W-media< 80-70) Pr(0<Z< 0.2)

plotDistr(.x, dnorm(.x, mean=0, sd=1), 
          cdf=FALSE, xlab="x", 
          ylab="Densidade",
          regions=list(c(0, 0.2)), col=c('#0080C0', '#BEBEBE'), legend=FALSE)

pnorm(0.2) - pnorm(0) pnorm(0.2)-0.5 0.07925971

QQ plot

amostra_normal<-rnorm(100)

qqnorm(amostra_normal,col="red")
abline(a=0,b=1)

u=seq(0,3.09,by=0.01)  
p=pnorm(u)  
p=p-0.5  
m=matrix(p,ncol=10,byrow=TRUE)  
m=round(m,4)  
DT::datatable(m)

Aula - Distribuição Normal - 22/02

Roberta A. Bastos

27/02/2021

P(x>3) =

Pr(X > -1,46) =

QQ plot