##problem 3 a
plot(dexp, xlim = c(0, 3))

##problem 3 b
plot(hist(rexp(100)))

##problem 4 b
Means <- apply(matrix(rexp(100 * 20), 100), 1, mean)
plot(hist(10 * (Means - 1)))

##problem 7 a
dat <- rnorm(100)
plot(hist(dat))
mean(dat) + c(-1, +1) * qnorm(.975) * sd(dat) / 10

##problem 8 a
ciFunc <- function(x, alpha = .05){
  mean(x) + c(-1, +1) * qt(1 - alpha / 2, df = length(x) - 1) * sd(x) / sqrt(length(x))
}

dat <- matrix(rnorm(20 * 9, mean = 69, sd = 3), 20)
cis <- t(apply(dat, 1, ciFunc))
mean(cis[,1] < 69 & cis[,2] > 69)

##problem 8 e
mean(cis[,1] < 69 & cis[,2] > 69)
mean(cis[,1] < 70 & cis[,2] > 70)
mean(cis[,1] < 72 & cis[,2] > 72)

##problem 8 h
Means <- apply(dat, 1, mean)
stem(Means)
var(Means)
##theoretical variance is sigma^2 / n = 9 / 9 = 1