Redvar

#################################################################
###                  REDUCCION DE VARIANZA                    ###
#################################################################
### Ejemplo de Clase:
rm(list = ls())
n=1000
U=vector()
X=vector()
for (i in 1:n)
{U[i]=runif(1,0,2)
 X[i]=exp(U[i])
};X
integral1=mean(X)
var1=var(X)/1000

n= 1000
X1 = vector()
U1 = vector()
U2 = vector()
for(i in 1:n)
{U1[i]=runif(1,0,2)
 U2[i]=1-U1[i]
 X1[i]=(exp(U1[i]))+(exp(U2[i]))
};X1
integral2=mean(X1)
var2=var(X1)/1000
cbind(integral1, integral2)
cbind(var1, var2)


#######################################################################
###                        PRACTICA N°5                             ###
###  Variables Antiteticas / Variables Control  / Condicionamiento  ###
#######################################################################
### VARIABLES ANTITETICAS
### Pregunta 1:
rm(list = ls())
n <- 1000
X <- vector()
U <- vector()

for(i in 1:n)
{
  U[i]=runif(1,0,1)
  X[i]=(exp((U[i])^2)*(1+(exp(1-(2*U[i])))))/2
};X
integral1=mean(X)
var1=var(X)/1000

n <- 1000
X1 <- vector()
U1 <- vector()
U2 <- vector()

for(i in 1:n)
{
  U1[i]=runif(1,0,1)
  U2[i]=1-U1[i]
  X1[i]=((exp((U1[i])^2))+(exp((U2[i])^2)))/2
};X1
integral2=mean(X1)
var2=var(X1)/1000

cbind(integral1, integral2)
cbind(var1, var2)


### Pregunta 2:
n <- 1000
X <- vector()
U1 <- vector()
U2 <- vector()

for(i in 1:n)
{
  U1[i]=runif(1,0,1)
  U2[i]=runif(1,0,1)
  X[i]=exp((U1[i]+U2[i])^2)
};X
integral1=mean(X)
var1=var(X)/1000

n <- 1000
X1 <- vector()
U1 <- vector()
U2 <- vector()

for(i in 1:n)
{
  U1[i]=runif(1,0,1)
  U2[i]=1-U1[i]
  X1[i]=((exp((U1[i]+U2[i])^2))+(exp((1-(U1[i]+U2[i]))^2)))/2
};X1
integral2=mean(X1)
var2=var(X1)/1000

cbind(integral1, integral2)
cbind(var1, var2)


### Pregunta 3:
# Caso a:
rm(list = ls())
n <- 1000
X <- vector()
S <- 0
cont <- 0
for(i in 1:n)
{
  for(j in 1:5)
  {
    U=runif(1,0,1)
    X[j]=-logb(U)
    S=S+(j*X[j])
  }
  if(S>=21.6)
  {
    cont=cont+1
  }
  S=0
};X
cont

# Caso b y Caso c:
n <- 500
X <- vector()
Y <- vector()
cont1 <- 0
cont2 <- 0
S1 <- 0
S2 <- 0
for(i in 1:n)
{
  for(j in 1:5)
  {
    U=runif(1,0,1)
    X[j]=-logb(U)
    Y[j]=-logb(1-U)

    S1=S1+(j*X[j])
    S2=S2+(j*Y[j])
  }
  if(S1>=21.6)
  {
    cont1=cont1+1
  }
  if(S2>=21.6)
  {
    cont2=cont2+1
  }
  S1=0
  S2=0
};X;Y
contador=cont1+cont2;contador

### Pregunta 5:
# Caso a:
rm(list = ls())
Z <- vector()
X <- vector()
n <- 1000
for(i in 1:n)
{
  U=runif(12,0,1)
  Z[i]=sum(U)-6

  X[i]=((Z[i])^3)*(exp(Z[i]))

};X
media <- mean(X);media
vari <- var(X)/n;vari

# Caso b:
Longitud <- (2*(1.96)*sqrt(vari))/(sqrt(n));Longitud
IC_inferior <- media-((1.96)*sqrt(vari)/sqrt(n));IC_inferior
IC_superior <- media+((1.96)*sqrt(vari)/sqrt(n));IC_superior


### Pregunta 6:
n <- 500
X1 <- vector()
X2 <- vector()
for(i in 1: n)
{
  U=runif(1,0,1)
  X1[i]=-logb(U)
  X2[i]=-logb(1-U)
}
cor(X1,X2)
X=c(X1,X2)
length(X)
media=mean(X);media
vari=var(X);vari

### Pregunta 7:
# Caso a: X~U(0,1)
n <- 1000
X <- vector() ## Variable control
I <- vector()
med_X <- 0.5  ## Media de la Unif.

for(i in 1:n)
{
  X[i]=runif(1)

  if( X[i]<0.8)
  { I[i]=1 }
  else {I[i]=0 }
}

p=cor(I,X);p
c=-cov(X,I)/var(X);c
I_C=mean(I)+(c*(mean(X)-med_X));I_C
Var_IC=var(I)*(1-p^2);Var_IC
Por_Red <- (1-(Var_IC/var(I)));Por_Red

# Caso b: X~exp(1)
n <- 1000
X <- vector()  ## Variable control
u <- vector()
I <- vector()
Ic <- vector()
med_X <- 1

for(i in 1:n)
{
  u[i]=runif(1,0,1)
  X[i]=-logb(u[i])

  if( X[i]<0.8)
  { I[i]=1 }
  else {I[i]=0 }
}

p=cor(I,X); p; p*p
c=-cov(X,I)/var(I); c
Est=mean(I)+(c*(mean(X)-med_X)); Est
Var_Est=var(I)*(1-p^2); Var_Est
Por_Red <- (1-(Var_Est/var(I))); Por_Red


### Pregunta 9:
# Con el Uso de la Variable Control.
# theta=E(X)
n <- 100
X <- vector()
Y <- vector()
U <- vector()
med_Y <- 0.5

for(i in 1:n)
{
  U[i]=runif(1,0,1)
  Y[i]=exp((U[i])^2)

  X[i]=
}