Law of Large Numbers & CLT

Convergence of Sample Averages

Law of Large Numbers (LLN)

If $X_1, X_2, \ldots, X_n$ are i.i.d. with mean $\mu$, then the sample mean converges to $\mu$ in probability as $n \to \infty$:

$\bar{X}_n = \frac{1}{n}\sum_{i=1}^n X_i \xrightarrow{P} \mu$

The larger your sample, the closer the average is to the true mean.

Central Limit Theorem (CLT)

Even more remarkable: regardless of the original distribution, the standardized sample mean converges to $N(0,1)$:

$\frac{\bar{X}_n - \mu}{\sigma / \sqrt{n}} \xrightarrow{d} N(0, 1)$

This is why the normal distribution appears everywhere — it is the universal limiting shape of averages.

Box-Muller Transform

To generate $Z \sim N(0,1)$ from uniform samples $U_1, U_2 \sim \text{Uniform}(0,1)$:

$Z = \sqrt{-2 \ln U_1} \cdot \cos(2\pi U_2)$

import math, random

rng = random.Random(42)
u1, u2 = rng.random(), rng.random()
z = math.sqrt(-2 * math.log(u1)) * math.cos(2 * math.pi * u2)
# z is a single standard normal sample

Your Task

Implement sample_mean(n, mu, sigma, seed) using Box-Muller with random.Random(seed). Return the sample mean of $n$ draws from $N(\mu, \sigma^2)$, rounded to 2 decimal places.