Continuous Distributions

Uniform Distribution

$X \sim U(a, b)$

This indicates that all outcomes from interval $(a, b)$ are equally likely.

$f(x) = \frac{1}{b - a} \quad \forall a < x < b$ $f(x) = 0 \quad (otherwise)$ $F(x) = 0 \quad \forall a \leq x$ $F(x) = \frac{x - a}{b - a} \quad \forall a < x < b$ $F(x) = 1 \quad \forall x \leq b$ $E(X) = \int^b_a x \frac{1}{b - a}dx = [\frac{x^2}{2(b - a)}]^b_a = \frac{b^2 - a^2}{2(b - a)} = \frac{a + b}{2}$ $Var(X) = \int^b_a x^2 \frac{1}{b - a}dx - E(X)^2 = [\frac{x^3}{3(b - a)}]^b_a - E(X)^2 = \frac{b^3 - a^3}{3(b - a)} - E(X)^2$ $= \frac{(b - a)(b^2 + ab + a^2}{3(b - a)} - E(X)^2 = \frac{b^2 + ab + a^2}{3} - \frac{a^2 + b^2 + 2ab}{4}$ $= \frac{4b^2 + 4ab + 4a^2 - 3a^2 - 3b^2 - 6ab}{12} = \frac{a^2 - 2ab + b^2}{12} = \frac{(a - b)^2}{12}$

Possible Usage: as a starting point in simulations, if we can generate a random number in $U(0, 1)$, we can arbitrarily expand the range to almost any other uniform distribution