## Importance sampling: Cauchy tail probability P(X > 2)

library(lattice)

p1 <- function(n = 500) 
{
    mean(rcauchy(n) > 2)
}

p2 <- function(n = 500) 
{
    0.5 * mean(abs(rcauchy(n)) > 2)
}

p3 <- function(n = 500) 
{
    h <- function(x) 2 / (pi * (1 + x^2))
    0.5 - mean(h(runif(n, 0, 2)))
}

p4 <- function(n = 500) 
{
    h <- function(x) 1 / (2 * pi * (1 + x^2))
    mean(h(runif(n, 0, 0.5)))
}

pdf <- 
    make.groups(p1 = replicate(50, p1(500)),
                p2 = replicate(50, p2(500)),
                p3 = replicate(50, p3(500)),
                p4 = replicate(50, p4(500)))

if (!interactive()) png("cauchy_tail.png", width = 500, height = 400)

stripplot(which ~ data, data = pdf,
          jitter = TRUE,
          abline = list(v = pcauchy(2, lower.tail = FALSE)),
          xlab = "Estimate of P(Cauchy > 2) using m=500")

if (!interactive()) dev.off()