# Normal Distribution
n = 5
normData = replicate(100, {
  xbar <- mean(rnorm(n))
})
plot(density(normData), xlab = 'x', main = 'Sampling distribution of sample mean (normal)')
curve(dnorm(x, 0, sqrt(1/n)), add = T, col = 2)


# Exercise 
n = 5
unifData <- replicate(10000, {
  xbar <- mean(runif(n))
})
plot(density(unifData), xlab = 'x', main = 'Sampling distribution of sample mean (uniform)')
curve(dnorm(x, 0.5, sqrt(1/(12*n))), add = T, col = 2)


# pdf of chisq distribution
dev.new()
curve(dchisq(x, 9), from = 0, to = 10, col = 1, 
      xlab = "x", ylab = "f(x)", lwd = 2, ylim = c(0,0.5))
curve(dchisq(x, 6), from = 0, to = 10, col = 2, add = TRUE, lwd = 2)
curve(dchisq(x, 5), from = 0, to = 10, col = 3, add = TRUE, lwd = 2)
curve(dchisq(x, 4), from = 0, to = 10, col = 4, add = TRUE, lwd = 2)
curve(dchisq(x, 3), from = 0, to = 10, col = 5, add = TRUE, lwd = 2)
curve(dchisq(x, 2), from = 0, to = 10, col = 6, add = TRUE, lwd = 2)
curve(dchisq(x, 1), from = 0, to = 10, col = 7, add = TRUE, lwd = 2)
legend("topright", legend = c(paste0("df = ", c(1:5, 6, 9))),
       col = 7:1,
       lty = 1, lwd = 2)

# pdf of t distribution
dev.new()
curve(dt(x, 30), from = -5, to = 5, col = 2, 
      xlab = "x", ylab = "f(x)", lwd = 2)
curve(dt(x, 10), from = -5, to = 5, col = 3, add = TRUE, lwd = 2)
curve(dt(x, 5), from = -5, to = 5, col = 4, add = TRUE, lwd = 2)
curve(dt(x, 1), from = -5, to = 5, col = 5, add = TRUE, lwd = 2)
curve(dnorm(x), from = -5, to = 5, col = 1, add = TRUE, lwd = 2)
legend("topleft", legend = c(paste0("df = ", c(1, 5, 10, 30)), 'Normal'),
       col =5:1,
       lty = 1, lwd = 2)

# pdf of F distribution
curve(df(x, 1, 1), from = 0, to = 5, col = 1, 
      xlab = "x", ylab = "f(x)", lwd = 2, ylim = c(0, 2.2))
curve(df(x, 2, 1), from = 0, to = 5, col = 2, add = TRUE, lwd = 2)
curve(df(x, 5, 2), from = 0, to = 5, col = 3, add = TRUE, lwd = 2)
curve(df(x, 10, 1), from = 0, to = 5, col = 4, add = TRUE, lwd = 2)
curve(df(x, 100, 100), from = 0, to = 5, col = 5, add = TRUE, lwd = 2)
legend("topright", legend = paste0("df1 = ", c(1, 2, 5, 10, 100), ', ', "df2 = ", c(1, 1, 2, 1, 100)),
       col = 1:5,
       lty = 1, lwd = 2)