## Simulate the first time a random walk returns to 0

random_walk_0 <- function()
{
    generateX <- function() sample(c(-1, 1), 1)
    S <- generateX()
    count <- 1
    while (S != 0)
    {
        S <- S + generateX()
        count <- count + 1
    }
    count
}

C <- replicate(1000, random_walk_0())


heads2 <- function(p)
{
    X <- runif(1) > p
    Y <- runif(1) > p
    count <- 2
    while (X || Y) 
    {
        X <- Y
        Y <- runif(1) > p
        count <- count + 1
    }
    count
}

C <- replicate(100000, heads2(0.5) - 2)

table(C)

p.hat <- 1 / (1 + mean(C))

p.geom <- dgeom(0:20, prob = 0.2)
p.geom <- p.geom / sum(p.geom)

chisq.test(table(C)[1:21], p = p.geom)