rm(list=ls(all=TRUE))
                                        #Recorrido aleatorio
p=0.5
n=50
u=runif(n)
X=NULL
x=0
for(i in 1:n)
  {
    if(u[i]<=p) y=1 else y=-1
    x=x+y
    X=c(X,x)
  }
X

B=10000
X50=NULL
for(j in 1:B)
  {
    u=runif(n)
    x=0
    for(i in 1:n)
      {
        if(u[i]<=p) y=1 else y=-1
        x=x+y
      }
    X50=c(X50,x)
  }
hist(X50)
sX50=sd(X50)

library(KernSmooth)
h=dpik(X50)
zest=bkde(X50,bandwidth=h)
t=zest$x
fe=zest$y
f=dnorm(t,0,sX50)
plot(t,f,type="l",col="blue")
lines(t,fe,col="red")