The Normal Distribution

The normal (Gaussian) distribution is the single most important distribution in all of quantitative finance — log-returns are routinely modelled as normal, the Black-Scholes formula is built on a log-normal terminal stock price, and every linear factor model's residual is assumed normal. This note builds the normal PDF, shows the two-parameter family $\mathcal{N}(\mu,\sigma^2)$ is fully characterised by its mean and variance, establishes the standardisation trick that reduces every probability calculation to a single reference table (the standard normal $\Phi$), and proves the closure-under-linear-transformation property that makes the normal family so analytically convenient. The normalising constant $1/\sqrt{2\pi\sigma^2}$ and the distribution's moments are derived from scratch in the accompanying derivation sub-note, kept separate so this note stays at reading pace.

Learning objectives. By the end of this note you should be able to: