Normal Distribution

The Bell Curve

$X \sim N(\mu, \sigma^2)$ is the most important distribution in probability. Its PDF is:

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

$E[X] = \mu, \qquad \text{Var}(X) = \sigma^2$

Standard Normal

$Z \sim N(0, 1)$. Any normal can be standardized: $Z = (X - \mu)/\sigma$.

CDF via Error Function

The CDF has no closed form in elementary functions, but Python provides it via math.erf:

$\Phi(x) = P(Z \leq x) = \frac{1}{2}\left[1 + \text{erf}\left(\frac{x}{\sqrt{2}}\right)\right]$

import math

def normal_cdf(mu, sigma, x):
    return 0.5 * (1 + math.erf((x - mu) / (sigma * math.sqrt(2))))

def normal_pdf(mu, sigma, x):
    return (1 / (sigma * math.sqrt(2 * math.pi))) * math.exp(-0.5 * ((x - mu) / sigma)**2)

The 68-95-99.7 Rule

Why It's Everywhere

The Central Limit Theorem (next lesson) guarantees that the sum of many independent random variables converges to a normal distribution, regardless of the original distribution.

Your Task

Implement normal_full(mu, sigma, x) that prints CDF$(x)$ then PDF$(x)$ of $N(\mu, \sigma^2)$, each rounded to 4 decimal places.