Uniform and Normal — Section 6: Continuous Distributions

Two of the most commonly used continuous distributions.

The Uniform $(a, b)$ distribution puts equal density on the interval $[a, b]$ :

f(x) = \frac{1}{b - a}, \quad a \le x \le b

with mean $(a + b)/2$ and variance $(b - a)^2/12$ . It models "no information beyond a known range" and is the building block for inverse-CDF random sampling.

The Normal $N(\mu, \sigma^2)$ — the bell curve — has density

f(x) = \frac{1}{\sigma\sqrt{2\pi}} \exp\!\left(-\frac{(x - \mu)^2}{2\sigma^2}\right)

The Normal dominates statistics for several reasons. Sums of many small independent effects converge to Normal by the Central Limit Theorem. Sums of independent Normals are Normal. The tails decay extremely fast: $P(|X - \mu| > 3\sigma) \approx 0.0027$ .

In finance, log-returns are often modeled as Normal (which makes prices lognormal). Good enough for many applications, but real markets have heavier tails than Normal predicts — a gap that drives stress testing and tail-risk modeling.

Two of the most commonly used continuous distributions.

The Uniform $(a, b)$ distribution puts equal density on the interval $[a, b]$ :

f(x) = \frac{1}{b - a}, \quad a \le x \le b

with mean $(a + b)/2$ and variance $(b - a)^2/12$ . It models "no information beyond a known range" and is the building block for inverse-CDF random sampling.

The Normal $N(\mu, \sigma^2)$ — the bell curve — has density

f(x) = \frac{1}{\sigma\sqrt{2\pi}} \exp\!\left(-\frac{(x - \mu)^2}{2\sigma^2}\right)