Gaussian Distribution

The problem, from first principles

Most naturally occurring measurement noise, sensor error, rounding error, the sum of many small independent effects, piles up into the same recognizable shape: symmetric, peaked at the average, thinning out fast at the extremes. That shape is the Gaussian, and it is not a coincidence it shows up everywhere; the Central Limit Theorem explains exactly why.

Here you'll implement the Gaussian's probability density function directly from its formula, then generate actual Gaussian-distributed samples from nothing but uniform random numbers using the Box-Muller transform, the concrete answer to "where does randomness with this specific shape actually come from."

From theory to code

Implement gaussian_pdf(x, mu, sigma) against the closed-form density in Theory, then sample_gaussian(mu, sigma, n, uniform_draws), which turns pairs of uniform random draws into Gaussian samples via Box-Muller. The signatures and docstrings are already in the editor.

Constraints

• sigma > 0 (a zero or negative standard deviation is undefined).
• x may be a scalar or a NumPy array of any shape.
• uniform_draws are pre-generated floats in [0, 1), exactly 2 * n of them, so results are deterministic and checkable.
• Return samples as a flat array of length n.

Hints