[likelihood_ratio_process.md] Update np.random → Generator API

[Chihiro2000GitHub]

Summary

This PR migrates legacy NumPy random API usage in likelihood_ratio_process.md as part of QuantEcon/meta#299.

A single generator rng = np.random.default_rng() is defined once in the setup cell and reused throughout, passed explicitly to the simulation functions.

Details

• Defined one rng = np.random.default_rng() in the imports/setup cell and reused it across all main-text cells and simulation functions.
• Migrated the legacy calls: np.random.beta → rng.beta, np.random.rand(N) → rng.random(N), np.random.rand(N, T) → rng.random((N, T)), and np.random.choice(...) → rng.choice(...).
• Threaded rng as an explicit argument through the simulation functions and their wrappers (simulate_likelihood_ratios, simulate_sequences, protocol_1/protocol_2 and their compute_*/analyze_* wrappers, simulate_markov_chain, analyze_markov_chains, compute_markov_selection_error) and updated the call sites.
• No fixed seed was introduced; the original code did not seed, so simulation behaviour is unchanged.

Note for reviewers

• simulate_likelihood_ratios is decorated with @jit but has no parallel=True/prange, so rng is passed in as an explicit argument (rng.beta) rather than lifting the draws into a precomputed array. Verified locally that this compiles and runs under Numba 0.62.1.
• rng is defined once in the setup cell rather than at the first point of use, because it is shared across many far-apart sections; a single top-level generator seemed clearer and easier to locate than repeating default_rng() per block.
• mc.simulate inside simulate_markov_chain uses QuantEcon's own internal RNG and is not an np.random.* call, so it is left unchanged (out of scope for this migration).

Hi @mmcky and @HumphreyYang, I'd be grateful if you could take a look when you have time.

[github-actions]

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[mmcky]

✅ Translation sync completed (zh-cn)

Target repo: QuantEcon/lecture-python.zh-cn
Translation PR: QuantEcon/lecture-python.zh-cn#228
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• lectures/likelihood_ratio_process.md