Problem 12.4.Rmd

---
title: "R Notebook"
output: html_notebook
---

#12.4.2

```{r}
par(mfrow=c(1,3))

hist(replicate(1000, mean(rbinom(n=10, size = 1,  prob=1/2))), main = "N=10", xlab = "estimate of mean")
hist(replicate(1000, mean(rbinom(n=20, size = 1,  prob=1/2))), main = "N=20", xlab = "estimate of mean")
hist(replicate(1000, mean(rbinom(n=100, size = 1,  prob=1/2))), main = "N=100", xlab = "estimate of mean")

```



#12.4.3
```{r}

lerrors <- vector()
i <- 1
for (epsilon in seq(from = 0, to = 0.5, length.out = 10 )){
  lerrors[i] <- mean(replicate(1000, mean(rbinom(n=10, size = 1,  prob=1/2 + epsilon)) - 0.5))
  i <- i + 1
}

plot(seq(from = 0, to = 0.5, length.out = 10 ), lerrors, col = "blue")
lines(seq(from = 0, to = 0.5, length.out = 10 ), lerrors, col = "blue")
lerrors

```

```{r}
epsilon <- seq(from = 0, to = 0.5, length.out = 10 )
first_throw <- rbinom(n=1, size=1, prob = 1/2)
cat("\nFirst Throw: ",first_throw)
count <- 1
N <- 10
last_throw <- first_throw
ep <- 0.1
all_throws <- vector()
all_throws[count] <- first_throw

j <- 1
for (ep in epsilon) {
  count <- 1
  while (count < N){
    if(last_throw == 1){
      throw <- rbinom(n=1, size=1, prob = 1/2 + ep)
    }
    else{
      throw <- rbinom(n=1, size=1, prob = 1/2 - ep)
    }
    count <- count + 1
    all_throws[count] <- throw
    last_throw <- throw
  }
  
  lerrors [j] <- abs(mean(all_throws) - 0.5)
  j <- j + 1
  
}
plot(epsilon,lerrors)
lines(epsilon, lerrors)
```


```{r}
epsilon <- seq(from = 0, to = 0.5, length.out = 10 )

first_throw <- rbinom(n=1, size=1, prob = 1/2)
last_throw <- first_throw
cat(first_throw)
lerrors <- vector()
i <- 0
for (ep in epsilon){
  lerrors[i] <- replicate(1000, rbinom(n=100, size = 1, prob = (ifelse( last_throw==1, ep, -ep))))
  i <- i +1
}



```
```{r}
mean(replicate(1000, mean(rbinom(n=20, size = 1,  prob=1/2 + 0.1)))) - 0.5
```
```{r}
set.seed(42)

markov_coin <- function(n, prob_head = .5, eps = .1) {
  res <- vector("integer", n)
  for (s in seq(n)) {
    res[s] <- if (s == 1) {
      rbinom(1, 1, prob_head)
    } else {
      rbinom(1, 1, prob_head + ifelse(res[s-1] == 1, eps, -eps))
    }
  }
  res
}

epsilon <- seq(0, .5, length.out = 10)
svec <- c(10, 20, 100)

res <- matrix(nrow = 10, ncol = 3)
for (s in seq_along(svec)) {
  for (eps in seq_along(epsilon)) {
    res[eps, s] <- sd(replicate(1000, mean(markov_coin(svec[[s]], .5, epsilon[eps])))) 
  }
}

plot(epsilon, res[,1], "b", col="blue", xlab = "epsilon", ylab = "error", ylim = c(0, .5))
lines(epsilon, res[,2], "b", col="orange")
lines(epsilon, res[,3], "b", col="green")
```