## Averages from exponentials

Exponentially distributed RVs donâ€™t look very normal:

• no negative values,
• heavy right tail.
y <- seq(0, 5, length=1000)
plot(y, dexp(y), type="l", lwd=2, main="Exp(1) density")

• This exponential has a mean (and variance) of 1.

First consider averages from a thousand samples of size 2.

y2 <- colMeans(matrix(rexp(2*1000), ncol=1000))
hist(y2, freq=FALSE, xlab="averages of 1000 samples of size 2")
lines(y, dnorm(y, 1, 1/sqrt(2)), col="red")
legend("topright", "AN", lty=1, col=2, bty="n")

Now averages from a thousand samples of size 5.

y2 <- colMeans(matrix(rexp(5*1000), ncol=1000))
hist(y2, freq=FALSE, xlab="averages of 1000 samples of size 5")
lines(y, dnorm(y, 1, 1/sqrt(5)), col="red")
legend("topright", "AN", lty=1, col=2, bty="n")

Size 10.

y2 <- colMeans(matrix(rexp(10*1000), ncol=1000))
hist(y2, freq=FALSE, xlab="averages of 1000 samples of size 10")
lines(y, dnorm(y, 1, 1/sqrt(10)), col="red")
legend("topright", "AN", lty=1, col=2, bty="n")

Size 100.

y2 <- colMeans(matrix(rexp(100*1000), ncol=1000))
hist(y2, freq=FALSE, xlab="averages of 1000 samples of size 100")
lines(y, dnorm(y, 1, 1/sqrt(100)), col="red")
legend("topright", "AN", lty=1, col=2, bty="n")

Size 1000.

y2 <- colMeans(matrix(rexp(1000*1000), ncol=1000))
hist(y2, freq=FALSE, xlab="averages of 1000 samples of size 1000")
lines(y, dnorm(y, 1, 1/sqrt(1000)), col="red")
legend("topright", "AN", lty=1, col=2, bty="n")