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

[Chihiro2000GitHub]

Summary

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

The lecture used the module-level np.random.seed / @jit set_seed() idiom for reproducibility. It is replaced with an explicit seeded generator rng = np.random.default_rng(142857) that is passed into the simulation functions.

Details

• Replaced the np.random.seed(142857) / @jit def set_seed() mechanism with rng = np.random.default_rng(142857). The lecture reset the seed at two points (setup cell and before the long-horizon paths), so rng is (re)created at both points to preserve that behaviour.
• Migrated np.random.beta → rng.beta and np.random.rand → rng.random, and threaded rng as an explicit argument through simulate, draw_lottery, draw_lottery_MC, simulate_mixed, and plot_π_seq, updating all call sites.
• Kept the existing fixed seed (142857); the original lecture already seeded for reproducibility, so no new seed was introduced.

Note for reviewers

• π_lim (used with np.vectorize): π_lim is wrapped by π_lim_v = np.vectorize(π_lim) and calls simulate_mixed. Adding rng as a positional argument would make np.vectorize try to broadcast it, so I left π_lim's signature unchanged and let it reference the shared module-level rng inside. This reuses the same single generator, just via closure rather than an argument. The explicit-argument alternative would be:

  def π_lim(α, rng, T=5000, π_0=0.4):
      ...
      l_arr = simulate_mixed(α, rng, T, N=1)[0]
      ...
  π_lim_v = np.vectorize(π_lim, excluded=['rng'])
  # called as π_lim_v(α_arr_x, rng=rng)

  Happy to switch to that if you prefer keeping rng explicit everywhere.
• Prose at "method 1": the pseudo-code bullet previously said "use the numpy.random.choice function to flip an unfair coin". I updated it to "use the rng.choice method …" to match the Generator API. Note the actual draw_lottery code implements the coin flip with rng.random() <= p rather than rng.choice, so this is a conceptual description — let me know if you'd rather word it differently.
• x=0.5 convergence figure: under the new generator the realized 200-step path in the "three different prior beliefs" figure (T_mix = 200) converges less visibly to x=0.5 than before, though the text says the posterior means "eventually converge". The theoretical statement still holds; if you'd like the figure to match more tightly, T_mix could be increased. I left it unchanged to keep this PR focused on the RNG migration.
• The @jit functions (simulate, draw_lottery) have no parallel=True/prange, so rng is passed in explicitly (Case 2). The numpyro/JAX MCMC uses jax.random.PRNGKey, which is not an np.random.* call and is left unchanged. Verified locally that the notebook compiles and runs under Numba 0.62.1.

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

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[jstac]

Many thanks @Chihiro2000GitHub , nice work. There is, however, an issue of overlapping PRs. I'll write separately to explain.