13  Exercises — Probability Track

End-of-chapter exercises that span the probability series. Each one asks for a small simulation or an analytic check that exercises the chain Bernoulli → Binomial → Poisson → Normal → CLT.

13.1 Exercise 1 — Binomial → Poisson convergence (quantitative)

For \lambda = 50, compute the maximum absolute difference between \text{Binomial}(n, \lambda/n) and \text{Poisson}(\lambda) PMFs for n \in \{50, 100, 500, 1000, 5000\}. Plot max difference vs n on a log–log scale. What is the convergence rate?

Hint: use scipy.stats.binom.pmf and scipy.stats.poisson.pmf; evaluate over k \in [0, 100].

Expected: approximately O(1/n) convergence — the slope on log– log axes should be near -1.

13.1.1 Steps

  1. For each n, compute both PMFs over the same k range.
  2. Take the maximum absolute difference across all k.
  3. Plot results on a log–log scale.
  4. Fit a line to estimate the slope.

13.2 Exercise 2 — Variance-stabilising transform (Anscombe)

The Poisson distribution has signal-dependent noise (\sigma = \sqrt\lambda). The Anscombe transform f(x) = 2\sqrt{x + 3/8} approximately stabilises the variance, making it constant regardless of \lambda. This transform is used in fluorescence microscopy and astronomical imaging before applying Gaussian denoisers — it converts Poisson data into a form where standard Gaussian denoising (NLM, BM3D, Wiener) is valid.

13.2.1 Steps

  1. Generate Poisson samples for \lambda \in \{5, 50, 500\} (10 000 samples each).
  2. Apply the Anscombe transform to each set.
  3. Compute the variance before and after.
  4. Plot histograms showing how the transform makes all three distributions have similar spread.

Expected: after the Anscombe transform, the variance should be approximately 1 for all \lambda values.

13.3 Exercise 3 — Build your own noise budget

You are building an image denoising pipeline for industrial inspection. Given:

  • Scene illumination: 200 photons/µm²/exposure
  • Photosite pitch: 3.45 µm (Basler acA1920-40gm)
  • Quantum efficiency: 0.65
  • Read noise: 4 electrons
  • Dark current: 2 electrons (cooled sensor)
  • Full well: 11 000 electrons
  • Bit depth: 12

13.3.1 Steps

  1. Compute the expected electron count per pixel.
  2. Compute each noise source’s contribution (variance).
  3. Compute the total noise and SNR at different signal levels.
  4. Determine where Gaussian denoising is valid vs where Anscombe pre-processing is needed (hint: at what signal level does Poisson ≈ Normal?).
  5. Simulate 10 000 exposures and verify your calculations.

Hint: photosite area = pitch²; expected electrons = flux × area × QE. Gaussian denoising is reasonable once \lambda > 20.

Expected: a noise-budget table, an SNR-vs-signal plot, and a simulated histogram that matches theory.

13.4 Exercise 4 — CLT convergence vs skewness

The CLT convergence rate depends on the skewness of the original distribution. Distributions with higher skewness need more samples to converge.

13.4.1 Steps

  1. Generate samples from Uniform(0, 1), Exponential(1), and Pareto(2, 1) — three distributions with increasing skewness.
  2. For each, compute the standardised sum of n samples for n \in \{1, 2, 5, 10, 30, 100\}.
  3. Measure the KS distance from the standard normal: scipy.stats.kstest(z, "norm").
  4. Plot KS distance vs n for all three on log–log axes.

Expected: higher-skewness distributions need more terms. Pareto slowest, Uniform fastest.

13.5 Exercise 5 — Variance equals mean (Poisson)

For \lambda \in \{1, 5, 20, 100, 500\}, draw 10 000 Poisson samples each and compute the sample mean and sample variance. Verify that they’re equal to within Monte-Carlo error. Plot both as a function of \lambda on a log–log scale.

13.6 Exercise 6 — The 1/\sqrt n noise reduction law

Generate n_{\text{trials}} = 10\,000 traces of length T = 1000 from Uniform(-1, 1). For each trace, compute the moving average with window length w \in \{1, 4, 16, 64, 256\}. For each w, compute the std of the smoothed signal. Plot std vs w on log–log axes; the slope should be -1/2.

13.7 Exercise 7 — Fixing a non-Bernoulli scenario

For each scenario, identify which of the three Bernoulli conditions (two outcomes / independence / constant p) fails and which distribution is appropriate instead:

  1. Drawing 5 cards from a deck without replacement, counting hearts.

  2. Tomorrow’s weather, given today’s.

  3. Whether a server is up at one-minute intervals throughout the day.

  4. Logging whether each photo in a benchmark is correctly classified by a fixed model.