Capítulo 7 Distribuicoes Multivariadas

S = rolldie(2, makespace = TRUE)
S = addrv(S, FUN = max, invars = c("X1", "X2"), name = "U")
S = addrv(S, FUN = sum, invars = c("X1", "X2"), name = "V")
head(S)

##   X1 X2 U V      probs
## 1  1  1 1 2 0.02777778
## 2  2  1 2 3 0.02777778
## 3  3  1 3 4 0.02777778
## 4  4  1 4 5 0.02777778
## 5  5  1 5 6 0.02777778
## 6  6  1 6 7 0.02777778

UV <- marginal(S, vars = c("U", "V"))
head(UV)

##   U V      probs
## 1 1 2 0.02777778
## 2 2 3 0.05555556
## 3 2 4 0.02777778
## 4 3 4 0.05555556
## 5 3 5 0.05555556
## 6 4 5 0.05555556

xtabs(round(probs, 3) ~ U + V, data = UV)

##    V
## U       2     3     4     5     6     7     8     9    10    11    12
##   1 0.028 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
##   2 0.000 0.056 0.028 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
##   3 0.000 0.000 0.056 0.056 0.028 0.000 0.000 0.000 0.000 0.000 0.000
##   4 0.000 0.000 0.000 0.056 0.056 0.056 0.028 0.000 0.000 0.000 0.000
##   5 0.000 0.000 0.000 0.000 0.056 0.056 0.056 0.056 0.028 0.000 0.000
##   6 0.000 0.000 0.000 0.000 0.000 0.056 0.056 0.056 0.056 0.056 0.028

marginal(UV, vars = "U")

##   U      probs
## 1 1 0.02777778
## 2 2 0.08333333
## 3 3 0.13888889
## 4 4 0.19444444
## 5 5 0.25000000
## 6 6 0.30555556

marginal(UV, vars = "V")

##     V      probs
## 1   2 0.02777778
## 2   3 0.05555556
## 3   4 0.08333333
## 4   5 0.11111111
## 5   6 0.13888889
## 6   7 0.16666667
## 7   8 0.13888889
## 8   9 0.11111111
## 9  10 0.08333333
## 10 11 0.05555556
## 11 12 0.02777778

temp <- xtabs(probs ~ U + V, data = UV)
rowSums(temp)

##          1          2          3          4          5          6 
## 0.02777778 0.08333333 0.13888889 0.19444444 0.25000000 0.30555556

colSums(temp)

##          2          3          4          5          6          7          8 
## 0.02777778 0.05555556 0.08333333 0.11111111 0.13888889 0.16666667 0.13888889 
##          9         10         11         12 
## 0.11111111 0.08333333 0.05555556 0.02777778

** Esperança Conjunta e Marginal **

Eu <- sum(S$U * S$probs)
Ev <- sum(S$V * S$probs)
Euv <- sum(S$U * S$V * S$probs)

Eu

## [1] 4.472222

Ev

## [1] 7

Euv

## [1] 34.22222

** Covariância e Correlação **

Euv - Eu * Ev

## [1] 2.916667

7.1 Distribuição condicional

x = c(rep(0,500),rep(1,500))
y = c(rnorm(500),rnorm(500,3))

plot(density(y))

plot(density(y[x==0]), ylim = c(0,0.5), xlim = c(-3,8))
lines(density(y[x==1]),col = "red")

mean(y)

## [1] 1.484076

mean(y[x==0])

## [1] 0.001430795

mean(y[x==1])

## [1] 2.966721

var(y)

## [1] 3.178244

var(y[x==0])

## [1] 0.8957608

var(y[x==1])

## [1] 1.061813

x = c(rep(0,500),rep(1,500))
y = c(rnorm(500,1,2),rnorm(500,1,5))

plot(density(y[x==0]), ylim = c(0,0.3), xlim = c(-10,10))
lines(density(y[x==1]),col = "red")

mean(y)

## [1] 0.888861

mean(y[x==0])

## [1] 0.9368569

mean(y[x==1])

## [1] 0.840865

var(y)

## [1] 14.24852

var(y[x==0])

## [1] 4.089858

var(y[x==1])

## [1] 24.43111

x = rnorm(1000)
y = rnorm(1000)

cor(x,y)

## [1] -0.03529856

plot(x,y)

x = rnorm(1000)
y = 2 + 0.5*x + rnorm(1000)

cor(x,y)

## [1] 0.4195284

plot(x,y)

x = rnorm(1000)
y = x^2 + rnorm(1000)

cor(x,y)

## [1] 0.02173687

plot(x,y)