Bug: PRNG key reuse in autodiff lecture + unused import in numpy_vs_numba_vs_jax

[mmcky]

Summary

Two minor issues found during translation review of the zh-cn repo (via Copilot code review on QuantEcon/lecture-python-programming.zh-cn#19):

1. PRNG key reuse in lectures/autodiff.md

In the gradient descent / simulated data section, the same key is used for both jax.random.uniform and jax.random.split:

x = jax.random.uniform(key, (n,))

α, β, σ = 0.5, 1.0, 0.1
key, subkey = jax.random.split(key)
ϵ = jax.random.normal(subkey, (n,))

This violates JAX's guidance that keys should be treated as single-use. key is consumed by uniform and then reused as input to split, which can lead to correlated draws.

Suggested fix:

key, x_key, eps_key = jax.random.split(key, 3)
x = jax.random.uniform(x_key, (n,))

α, β, σ = 0.5, 1.0, 0.1
ϵ = jax.random.normal(eps_key, (n,))

2. Unused random import in lectures/numpy_vs_numba_vs_jax.md — resolved

Resolved — verified against origin/main on 2026-08-03: numpy_vs_numba_vs_jax.md no longer imports random. Only item 1 above remains live.

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Found by GitHub Copilot code review during automated translation sync. (Body updated 2026-08-04: item 2 struck as already resolved.)