The Normal Distribution
The normal (or Gaussian) distribution is the bell-shaped
curve at the centre of probability. We write X \sim N(\mu, \sigma^2)
to say that X has mean
\mu and variance
\sigma^2. Its density — the height of the curve at each point — is
f(x) = \frac{1}{\sigma\sqrt{2\pi}}\,\exp\!\left(-\frac{(x-\mu)^2}{2\sigma^2}\right).
That \exp of a negative square (see the
exponential function) is what makes
the curve fall away smoothly on both sides. It is symmetric about
\mu: the mean is also the peak and the median. The number
\sigma sets the width — small
\sigma gives a tall, narrow spike; large
\sigma a low, broad mound.
Standardising: the one curve behind them all
Every normal is a shifted, stretched copy of a single master curve. The
standard normal Z \sim N(0,1) has mean
0 and variance 1; its cumulative
distribution function is written \Phi(z) = \mathbb{P}(Z \le z) — the
area under the bell to the left of z. There is no elementary formula
for \Phi, which is exactly why it gets its own symbol (and why it
shows up inside the Black–Scholes formula).
To turn any X \sim N(\mu, \sigma^2) into a standard normal, subtract
the mean and divide by the standard deviation:
Z = \frac{X - \mu}{\sigma} \sim N(0,1).
This standardisation measures X in units of
\sigma away from the mean, so a single table of
\Phi answers questions about every normal.
Shape the bell
Slide \mu to move the curve left and right, and
\sigma to make it wider or narrower. The total area
under the curve is always 1, so a narrower bell must grow taller. The
faint curve is the fixed standard normal N(0,1) for comparison.
The famous 68–95–99.7 rule reads straight off this picture: about
68\% of the area lies within 1\sigma of
\mu, about 95\% within
2\sigma, and about 99.7\% within
3\sigma.
Why finance cares
Over a short period a stock's log-return
\ln(S_{t+\Delta}/S_t) is modelled as normal. Returns are roughly
symmetric, they add up cleanly over time (a sum of independent normals is again normal), and the
normal is the limit the central limit theorem hands us when many small shocks
accumulate. That single modelling choice — normal log-returns — is the seed that grows into the
lognormal
model of the price itself.