Fix assorted issues in prob_meaning lecture

[jstac]

Summary

Addresses #877 — a comprehensive revision of lectures/prob_meaning.md.

Mechanical fixes:

• Typos (probabililty ×6, statististian, etc.), doubled words ("to to", "with with")
• Wrong variance formula (extra factor of n), swapped upper/lower bounds
• Notation: $P_{k,i}$ → $\rho_{k,i}$, subject-verb agreement, LaTeX * → \cdot
• compare() now includes $k=0$, log(I) range text matches code
• Imprecise wording: $f_k^I$ approximates $\text{Prob}(X=k|\theta)$, not $\theta$
• Use IID instead of i.i.d., **bold** instead of __bold__

Code quality:

• PEP 8 naming (frequentist → Frequentist, Bay_stat → bayes)
• Clearer variable names (ii → post/i, kk → k, K → head_counts, etc.)
• Replaced prettytable dependency with pandas DataFrame
• Reproducible output via np.random.default_rng() seeds (modern NumPy API)
• Standardized plotting to use ax. methods instead of plt.
• Grid resolution bumped from 100 to 1000 points for smoother density plots
• f-string formatting with comma separators for plot labels
• Removed dead self.k assignment

Content restructuring:

• Derived the $n$-step posterior $\text{Beta}(\alpha + k, \beta + n - k)$ before the exercise, replacing the duplicated derivation that appeared after
• Added back-reference to prob_matrix lecture for Bayes' Law
• Introduced $p(\theta)$ (lowercase) as density notation, noted it is a density
• Broke up the crowded aligned equation in part (b) solution into three display equations with explanatory text
• Removed repeated question text from solution headers (QuantEcon convention)
• Renamed quantile variables $p_1, p_2$ → $q_1, q_2$ to avoid clash with density notation
• Standardized $N$ → $n$ throughout
• Cross-references now use actual lecture titles for better PDF rendering
• Simplified plot legends and reduced clutter (removed n = 30, 70, 300, 500)

Test plan

• Verify the lecture builds without errors
• Check that all code cells execute correctly
• Confirm plots render with improved grid resolution and reproducible seeds
• Verify cross-references resolve correctly

🤖 Generated with Claude Code

[jstac]

This PR addresses the issues catalogued in #877. The changes are split across three commits for easier review:

1. Mechanical fixes — typos, spelling, the wrong variance formula, swapped bounds, notation inconsistencies, and other small corrections.
2. Code quality — PEP 8 naming, clearer variable names, shorter docstrings, line length fixes, higher grid resolution for plots, and consistent use of .astype(int).
3. Content restructuring — the key pedagogical change. The $n$-step posterior formula $\text{Beta}(\alpha + k, \beta + n - k)$ is now derived before the exercise, so the solution code no longer relies on a result that hasn't been shown yet. The duplicated derivation that appeared after the exercise is replaced with a concise summary. A back-reference to the prob_matrix lecture is added for readers who need a refresher on Bayes' Law, and cross-references now use actual lecture titles for better PDF rendering.

[github-actions]

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Commit: 400d8ab

📚 Changed Lectures

• prob_meaning

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Build Info

• Workflow: Build Project [using jupyter-book]

[jstac]

Note: If QuantEcon/QuantEcon.manual#84 is merged (which standardizes on \mathbb{P} for probability notation), this PR will need a follow-up commit to replace \textrm{Prob}(...) with \mathbb{P}\{...\} throughout.

[jstac]

@thomassargent30

Please let me know if you would like any modifications to the PR. I'll aim to merge tomorrow but we can always walk things backwards.

I think these are all good changes.