Exercícios de Distribuição Normal

Com base nessa tabela, calcule as seguintes probabilidades:

1) P(X < 0) = 0,5

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="X", ylab="Densidade",regions=list(c(0,4)), col=c("#f0a3a3"), legend=FALSE)  

2) P(0 < X < 0,11)= 0,04380

.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(0,0.11)), col=c("#d4956c"), legend=FALSE)  

3) P(-2,2 < X < 0,12) = 0,53386

P(-2,2 < x < 0,12) = P(-2,2 < x < 0)+ P(0 < x < 0,12)

P(-2,2 < x < 0,12) = P(0 < x < 2,2) + P(0 < x < 0,12)

P(2,2 < x < 0,12) = 0,48610 + 0,04776

P(2,2 < x < 0,12) = 0,53386

.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(-2.2,0.12)), col=c("#e8eb67"), legend=FALSE)

plotDistr(.x, dnorm(.x, mean=0, sd=1), cdf=FALSE, xlab="X", 
          ylab="Densidade",regions=list(c(-2.2,0),c(0,0.12)), 
          col=c('#f0ed37','#9e9d3f'), legend=FALSE)

4) P(-1 < X < 2,1) = 0,82348

P(-1 < X < 2,1) = P(-1 < x < 0) + P( 0 < X < 2,1)

P(-1 < x < 2,1) = P(0 < x < 1) + P( 0 < X < 2,1)

P(-1 < x < 2,1) = 0,34134 + 0,48214

P(-1 < x < 2,1) = 0,82348

.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(-1,2.1)), col=c("#82c75a"), legend=FALSE)

plotDistr(.x, dnorm(.x, mean=0, sd=1), cdf=FALSE, xlab="X", 
          ylab="Densidade",regions=list(c(-1,0),c(0,2.1)), 
          col=c('#40c960','#306e3f'), legend=FALSE)

5) P(0 < X < 1,83) = 0,46638

.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(0,1.83)), col=c("#5ac7a3"), legend=FALSE)

6) P(-0,87 < X < 1,54) = 0,74607

P(-0,87 < X < 1,54) = P(-0,87 < x < 0) + P(0 < X < 1,54)

P(-0,87 < X < 1,54) = P(0 < x < 0,87) + P(0 < X < 1,54)

P(-0,87 < X < 1,54) = 0,30785 + 0,43822

P(-0,87 < X < 1,54) = 0,74607

.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(-0.87,1.54)), col=c("#5ddfe8"), legend=FALSE)

plotDistr(.x, dnorm(.x, mean=0, sd=1), cdf=FALSE, xlab="X", 
          ylab="Densidade",regions=list(c(-0.87,0),c(0,1.54)), 
          col=c('#54bfc7','#23858c'), legend=FALSE)

7) P(X = 1,54) = 0

Linha não tem área,sendo sempre zero.

Então o valor da probabilidade vai ser sempre zero quando x for igual a algum número.

8) P(x > 2,5) = 0,5 - 0,49379 = 0.00621

plotDistr(.x, dnorm(.x, mean=0, sd=1), cdf=FALSE, xlab="X", 
          ylab="Densidade",regions=list(c(0,2.5),c(2.5,4)), 
          col=c('#fa1b57','#850728'), legend=FALSE)

9) P(x > -2) = 0,47725 + 0,5 = 0,97725

plotDistr(.x, dnorm(.x, mean=0, sd=1), cdf=FALSE, xlab="X", 
          ylab="Densidade",regions=list(c(-2,0),c(0,4)), 
          col=c('#5e2a66','#b25ebf'), legend=FALSE)