Uniform random variables

• Definition: Let $X$ be a continuous uniform random variable defined in an interval $[a; b]$, then:
  • $f_X(x) = \dfrac{1}{b-a}$, if $x\in [a; b]$.
  • $f_X(x) = 0$, if $x\notin [a; b]$.
  • Notation: $X \sim Uniform(a, b)$
• Theorem: If $X \sim Uniform(a, b)$, then:
  • $\mathbb{E}[X] = \dfrac{a+b}{2}$.
  • $Var[X] = \dfrac{(b-a)^2}{12}$.

# Uniform(0, 1)
x <- seq(-1, 2, 0.01)

plot(
  x, dunif(x, 0, 1), type = 'l',
  main = "PDF", xlab = "x", ylab = ""
)

plot(
  x, punif(x, 0, 1), type = 'l',
  main = "CDF", xlab = "x", ylab = "",
  ylim = c(0,1)
)