5 Continuous Distributions

5.1 Uniform distributions

Here are the density, (cumulative) distribution function, survival function, and quantile function of the uniform distribution on $[0,1]$

x = seq(-0.2, 1.2, len = 1000)
u = seq(0, 1, len = 1000)
par(mfrow = c(2,2), mai = c(0.35, 0.35, 0.1, 0.1))
plot(x, dunif(x), type = "l", lwd = 2, ylab = "", xlab = "")
plot(x, punif(x), type = "l", lwd = 2, ylab = "", xlab = "")
plot(x, 1-punif(x), type = "l", lwd = 2, ylab = "", xlab = "")
plot(u, qunif(u), type = "l", lwd = 2, ylab = "", xlab = "")
abline(v = c(0, 1), lty = 2)

5.2 Normal distributions

Here are the density, (cumulative) distribution function, survival function, and quantile function of the standard normal distribution (mean 0, variance 1)

x = seq(-3, 3, len = 1000)
par(mfrow = c(2,2), mai = c(0.35, 0.35, 0.1, 0.1))
plot(x, dnorm(x), type = "l", lwd = 2, ylab = "", xlab = "")
plot(x, pnorm(x), type = "l", lwd = 2, ylab = "", xlab = "")
plot(x, 1-pnorm(x), type = "l", lwd = 2, ylab = "", xlab = "")
plot(u, qnorm(u), type = "l", lwd = 2, ylab = "", xlab = "")
abline(v = c(0, 1), lty = 2)

5.3 Exponential distributions

Here are the density, (cumulative) distribution function, survival function, and quantile function of the exponential distribution with rate 1

x = seq(-1, 5, len = 1000)
par(mfrow = c(2,2), mai = c(0.35, 0.35, 0.1, 0.1))
plot(x, dexp(x), type = "l", lwd = 2, ylab = "", xlab = "")
plot(x, pexp(x), type = "l", lwd = 2, ylab = "", xlab = "")
plot(x, 1-pexp(x), type = "l", lwd = 2, ylab = "", xlab = "")
plot(u, qexp(u), type = "l", lwd = 2, ylab = "", xlab = "")
abline(v = c(0, 1), lty = 2)

5.4 Normal approximation to the binomial

The de Moivre–Laplace theorem says that, as $n$ increases while $p$ remains fixed, the binomial distribution with parameters $(n, p)$ is well approximated by the normal distribution with same mean ( $= np$ ) and same variance ( $= n p (1-p)$ ). To verify this numerically, we fix $p = 0.10$ and vary $n$ in $\{10, 30, 100\}$ . We can see that the approximation is already very good when $n = 100$ .

par(mfrow = c(1, 3), mai = c(0.5, 0.5, 0.1, 0.1))
n = 10
p = 0.1
plot(0:n, dbinom(0:n, n, p), type = "h", lwd = 2, xlab = "", ylab = "")
curve(dnorm(x, n*p, sqrt(n*p*(1-p))), add = TRUE, lty = 2)

n = 30
p = 0.1
plot(0:n, dbinom(0:n, n, p), type = "h", lwd = 2, xlab = "", ylab = "")
curve(dnorm(x, n*p, sqrt(n*p*(1-p))), add = TRUE, lty = 2)

n = 100
p = 0.1
plot(0:n, dbinom(0:n, n, p), type = "h", lwd = 2, xlab = "", ylab = "")
curve(dnorm(x, n*p, sqrt(n*p*(1-p))), add = TRUE, lty = 2)