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coin.power <- function(a, p, n){
  c <-  qnorm(1 - a/2)  
  mu <- sqrt(n) * (2 * p - 1)   
  sigma <- sqrt(4 * p * (1 - p))
  
  less.than <- pnorm( -c, mean = mu, sd = sigma)
  greater.than <- 1 - pnorm(c, mean = mu, sd = sigma)
  power <- less.than + greater.than
  return(power)
}

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group <- c('treatment', 'control', 'refused')
n.children <- 1000 * c(200, 200, 340)
n.polio <- c(57, 142, 157)
rate <- n.polio/n.children
polio.data <- data.frame(group, n.children, n.polio, rate)
polio.data



treatment <- subset(polio.data, group == 'treatment')

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pnorm(10, mean = 12, sd = 2)



pnorm(250, mean = 300, sd = 10)



n <- 950:961
width <- 2 * qnorm(1 - 0.01/2) * 6/sqrt(n)

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%  I learned how to make these flowcharts by looking at some very helpful examples online:
%
%    http://www.texample.net/tikz/examples/simple-flow-chart/
%    http://www.texample.net/tikz/examples/borrowers-and-lenders/
%    http://www.texample.net/tikz/examples/entity-relationship-diagram/
%
%  For a long list of tikz examples, see http://www.texample.net/tikz/examples/		

\documentclass{beamer}  
\usepackage{amssymb, amsmath, amsthm} 

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ones <- rep(1, 10)
ones
cumsum(ones)
cumsum(1:10)



n <- 10000
sims <- rnorm(n, mean = 0, sd = 10)
running.mean <- cumsum(sims)/(1:n)

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#Function to draw a single Chi-squared random variable with degrees of freedom equal to df
my.rchisq <- function(df){
  
  #Draw df independent standard normals
  normal.sims <- rnorm(df)
  
  #Square them and sum the result
  chi.sims <- sum(normal.sims^2)

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x <- seq(from = -5, to = 5, by = 0.01)
y1 <- dnorm(x)
y2 <- dt(x, df = 1)
y <- cbind(y1, y2)
matplot(x, y, lty = 1, type = 'l', ylab = 'f(x)')

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x <- seq(from = -1, to = 1, by = 0.5)
y <- x^2
plot(x, y)



x <- seq(from = -1, to = 1, by = 0.1)
y <- x^2
plot(x, y)

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#This function simulates n independent, identically distributed Bernoulli Random Variables
rbern <- function(n, p){
  
  sims <- sample(0:1, size = n, replace = TRUE, prob = c(1-p, p))
  return(sims)
  
}

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marbles <- c('red', 'blue', 'green')
sample(x = marbles, size = 2, replace = FALSE)



sample(x = 1:10, size = 5, replace = FALSE)