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

[Chihiro2000GitHub]

Summary

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

Details

• Replaced np.random.seed(...) with rng = np.random.default_rng(...) throughout the lecture.
• Replaced np.random.randn, np.random.exponential, np.random.uniform, np.random.standard_t, and np.random.normal with the corresponding rng.* methods.
• In the Q-Q plots cell, rng from the preceding empirical CCDF cell (seed 13) is reused directly — the Q-Q plots compare data_normal against data_exp, data_logn, and data_pareto, all of which were generated with rng = np.random.default_rng(13), so reusing the same rng for data_normal keeps the comparison consistent.
• For scipy.stats calls (cauchy().rvs, pareto.rvs), random_state=rng is now passed explicitly, as confirmed in the scipy documentation.
• In the exercise ht_ex5 solution, the function signatures of pareto_rvs and tax_rev were updated to accept rng as an explicit argument. This is flagged for your judgment as it changes the function signatures.
• Exercise prose at ht_ex3 and ht_ex5 updated to reflect the new API.

Figures to review — please compare with the original rendered lecture:

• Fig. 22.6 (draws from the Cauchy distribution): The prose describes "a few sudden large deviations from the mean", but after the API migration the new figure shows deviations of noticeably smaller magnitude than the original. It may be worth considering whether a prose update or a seed/parameter adjustment is needed. I have not made any changes here and leave this to your judgment.

• Fig. 22.16 (Failure of the LLN): After the API migration, this figure could be interpreted as possibly converging if n were increased further, which may weaken the pedagogical point. It may be worth considering whether the seed or N should be adjusted. No changes made; leaving this to your judgment.

• Exercise 22.4 violin plot: The y-axis scale changed from 1e6 to 1e7 after the migration (likely due to more extreme draws in the Pareto case), causing the plot to appear more compressed than before. The prose remains accurate, but the figure may be less readable. Leaving this to your judgment.

Something I noticed (not fixed in this PR)

While working on this file, I noticed some inconsistencies in the exercise numbering — just flagging them here in case they are useful. These were already present before this PR and are unrelated to the current changes. Exercise 22.1 does not exist on the rendered page (exercises start at 22.2). Exercise 22.3 (ht_ex3) says "Repeat exercise 1", but the exercise it refers to is rendered as Exercise 22.2, not 22.1. Additionally, the exercise rendered as 22.4 corresponds to the label ht_ex5 in the source, creating a mismatch between displayed numbers and internal labels. I have not made any changes here.

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

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