Fix assorted issues in prob_meaning lecture (#878)

* Fix typos, spelling, notation errors, and swapped bounds in prob_meaning

- Fix spelling: probabilties, probabililty (x6), statististian
- Fix doubled word: "to to help"
- Fix variance formula: remove erroneous factor of n (rho is Bernoulli, not binomial)
- Fix notation: P_{k,i} → rho_{k,i} to match definition
- Fix subject-verb agreement: "means converges" → "mean converges"
- Fix swapped upper/lower bounds in part (e) ppf calls
- Fix compare() to include k=0
- Fix LaTeX: replace * with \cdot for multiplication
- Fix log(I) range: text said 2 to 7, code has 2 to 6
- Fix imprecise wording: f_k^I approximates Prob(X=k|θ), not θ
- Clarify vague exercise pm_ex1 part 3

Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>

* Improve code quality: naming, PEP 8, line length, grid resolution

- Rename class frequentist → Frequentist (PEP 8)
- Rename Bay_stat → bayes (snake_case for instances)
- Rename ii → i/post, num → n_obs, num_list → n_obs_list,
  kk → k, K → head_counts, comp → table, step_num → n_obs,
  npt → n_thetas, nn → n_ns, nI → n_Is
- Replace (sample <= θ) * 1 with .astype(int) for consistency
- Shorten docstrings to fit within 80 characters
- Break long code lines (plot calls, list comprehensions, titles)
- Increase θ grid from 100 to 1000 points for smoother density plots
- Use f-strings with comma formatting for plot labels

Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>

* Restructure Bayesian section: derive posterior before exercise

- Add back-reference to prob_matrix lecture for Bayes' Law intro
- Derive the n-step posterior Beta(α+k, β+n-k) before the exercise,
  so the exercise solution code no longer precedes its own derivation
- Replace the duplicated derivation after the exercise with a concise
  summary referencing the formula above
- Remove duplicate "Now pretend..." sentence before part (c)
- Replace "this quantecon lecture" cross-references with actual titles
  for better PDF rendering

Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>

* Replace prettytable with pandas DataFrame in compare()

Drop the prettytable dependency — pandas is already imported and
renders nicely in Jupyter notebooks. The compare() method now returns
a DataFrame instead of printing a PrettyTable.

Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>

* Fix exercise pm_ex2 parts (a) and (b) to match their solutions

The question asked for the likelihood of "a sample of length n from
a binomial" but the solution gave the single-flip Bernoulli case.
Reword both the questions and solution headers so parts (a) and (b)
are explicitly about a single coin flip. The general n-step case is
already derived in the lecture text before the exercise.

Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>

* Add reproducible random seeds using modern NumPy API

Both classes now accept an rng parameter and use rng.random()
instead of np.random.rand(). Each code cell passes a seeded
np.random.default_rng() for reproducible output across builds.

Also remove "typically" from the hump-shape sentence, since with
fixed seeds the behavior is deterministic.

Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>

* Clean up posterior PDF plots

- Rename title from "P.D.F" to "PDF"
- Simplify legend labels: "n = 0 (prior)", "n = 1", etc.
- Remove n = 30, 70, 300, 500 from observation list to reduce
  clutter in the first PDF plot

Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>

* Remove repeated question text from exercise solutions

Solutions for parts (c)-(h) no longer duplicate the question text
as a header — they just use the part label. This follows the
QuantEcon convention.

Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>

* Use p(θ) for density notation and break up solution derivation

- Change P(θ) to p(θ) throughout and note it is a density
- Replace the single aligned equation block in the solution for
  part (b) with three separate display equations, each introduced
  by explanatory text (Bayes' Law, substitution, collecting powers)

Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>

* Final review polish: notation, plotting style, consistency

- Use **bold** instead of __bold__ for binomial distribution
- Use IID instead of i.i.d.
- Fix double "with" in exercise (c) wording
- Rename quantile variables from p_1/p_2 to q_1/q_2 to avoid
  clash with p(θ) density notation
- Fix "means and variances statistics" → "mean and standard deviation"
- Standardize N → n in post-exercise text to match pre-exercise
- Update "exceeds 500" → "exceeds 1000" to match revised n_obs_list
- Standardize frequentist plots to use ax. methods instead of plt.
- Remove dead self.k assignment in Frequentist.binomial()

Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>

* Adopt new style guide notation conventions

Replace \textrm{Prob}(...) with \mathbb{P}{...} and E[...] with
\mathbb{E}[...] following QuantEcon.manual#84.

Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>

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Co-authored-by: Claude Opus 4.7 (1M context) <noreply@anthropic.com>

Author: 2 people