Chapter 2

Chapter 2: Core Distributions for MLOps

Essential probability distributions in production ML: Normal, Student-t, Bernoulli/Binomial, Poisson, Exponential, and when to use each

Distributions

Normal (Gaussian, "z")

What it models

  • Measurement noise, aggregated effects, residual errors.
  • The “default” approximation for averages due to CLT.

Where it appears in ML systems

  • Regression residual assumptions (even if imperfect).
  • Error distributions for monitoring (e.g., prediction error aggregated over time).
  • Many statistical tests/intervals assume approximate normality of an estimator.

How to recognize

  • Symmetric, bell-shaped; tails not too wild.
  • Hist of residuals roughly symmetric; Q-Q plot close to line.

Practical notes

  • Normality of raw data is often false; normality of mean / estimator can still hold.
  • If you see heavy tails → switch to robust stats or transform.

Student-t

What it models

  • Uncertainty in the mean when variance is unknown and sample size is small.
  • Like Normal but heavier tails (more conservative).

Where it appears

  • A/B tests on means (AOV, time-on-site) when N is not huge.
  • Confidence intervals for mean with unknown σ.

Recognize

  • Same shape as Normal but with heavier tails; converges to Normal as df↑.

Rule

  • If you’re doing “mean difference” with unknown variance, you’re in t-world (or Welch’s t if variances differ).

Bernoulli / Binomial

Bernoulli

  • Single trial 0/1 event: click, convert, success/failure label.

Binomial

  • Sum of Bernoullis over (n): #conversions out of n visits.

Where it appears

  • CTR, conversion rate, precision@k (binary outcomes), label prevalence.
  • Online monitoring of success rates.

Recognize

  • Data is 0/1 or counts of successes.
  • Variance tied to mean: (p(1-p)).

What you do

  • CIs for proportions (Wilson often beats naive Wald).
  • Hypothesis tests: z-test / chi-square / Fisher exact.
  • Bayesian: Beta posterior over (p) (next section).

Poisson (counts per interval)

What it models

  • Counts of events in fixed time/space when events happen independently at a constant rate.

Where it appears

  • Request counts per second/minute, errors per minute, clicks per hour.
  • Incident rates.

Recognize

  • Nonnegative integers; often variance ≈ mean (key property).
  • Inter-arrival times ~ Exponential (dual relationship).

What breaks it

  • Over-dispersion (variance >> mean) from burstiness, heterogeneity, seasonality.

What you do

  • If variance >> mean → Negative Binomial (or mixture models).
  • For monitoring: often use Poisson process intuition + change-point detection.

Exponential / Gamma

Exponential

What it models

  • Waiting time between Poisson events (“time to next request/click”).

Where it appears

  • Inter-arrival times, simplified failure times.

Recognize

  • Positive continuous; memoryless property.

Gamma

What it models

  • Sum of exponentials (waiting time to the k-th event).
  • Positive skewed continuous values.

Where it appears

  • Latency/service time modeling (sometimes), durations, time-to-complete tasks.
  • Bayesian conjugate prior for Poisson rate (Gamma–Poisson).

Recognize

  • Positive, right-skewed; can be flexible (shape controls skew).

Heuristic

  • If it’s a positive skewed duration and lognormal isn’t perfect, Gamma is often a good candidate.

Beta

What it models

  • A distribution over a probability (p\in[0,1]).

Where it appears

  • Bayesian modeling of CTR/conversion.
  • Uncertainty over rates, especially with small sample sizes.

Why it’s loved

  • Conjugate with Bernoulli/Binomial → posterior update is simple:

    • prior Beta(α,β) + data (successes, failures) → Beta(α+succ, β+fail)

Recognize

  • Values in [0,1], can be uniform, U-shaped, skewed, peaked depending on α,β.

Heuristic

  • When you want “probability with uncertainty” for rates, Beta is the natural language.

Dirichlet

What it models

  • Distribution over a probability vector (multiclass proportions), generalization of Beta.

Where it appears

  • Class distribution drift monitoring (multinomial outcomes).
  • Bayesian smoothing for categorical frequencies (tokens/categories).

Practical use

  • Helps avoid zero-probabilities (smoothing) when computing divergences like KL/JS.
  • Conjugate prior for Multinomial.

Heuristic

  • If you have K categories and want a principled “prior + counts → posterior proportions,” Dirichlet.

Lognormal

What it models

  • Positive quantities where the log is approximately normal.
  • Multiplicative processes (many small multiplicative factors).

Where it appears

  • Latency, response sizes, spend/AOV, time-on-task.

Recognize

  • Positive, strong right tail; log(x) looks more normal.
  • Mean >> median (common sign).

What you do

  • Analyze on log scale (t-tests on log values often behave better).
  • Report medians/quantiles; use trimmed means for heavy tails.

Heuristic

  • For latency/spend: try log transform early.

Pareto / heavy-tailed families

What it models

  • “A few huge values dominate” (power-law behavior).

Where it appears

  • Spend/user, session durations, content popularity, tail latencies, long-tail item interactions.

Why it matters in MLOps

  • Means become unstable; variance may be enormous.
  • A/B tests and monitoring get noisy unless you use robust methods.

Recognize

  • Extreme outliers; log-log plots show heavy tail-ish; top 1% contributes a huge fraction.

What you do

  • Use robust stats: median, trimmed mean, winsorization.
  • Consider quantile-based metrics; bootstrap.
  • Sometimes model the tail separately (extreme value theory) if it’s core.

Heuristic

  • If one user can swing your metric, you’re in heavy-tail land.

Negative Binomial (over-dispersed counts)

What it models

  • Counts when variance > mean (Poisson is too “tight”).
  • Often: a Poisson rate that varies across users/time (Gamma–Poisson mixture).

Where it appears

  • Click counts per user, errors per service with burstiness, events per session.

Recognize

  • Over-dispersion: sample variance >> sample mean.
  • Many zeros + occasional big counts.

What you do

  • Use NB for modeling; for tests, use robust/permutation methods if assumptions shaky.

Heuristic

  • If Poisson underestimates spikes, switch to Negative Binomial.

Mixture models (multi-modal behavior)

What it models

  • Data generated by multiple latent subpopulations.

Where it appears

  • New vs returning users, bots vs humans, enterprise vs consumer traffic.
  • Latency: cache hit vs miss
  • CTR: different intent cohorts

Recognize

  • Histogram shows multiple peaks; metrics differ wildly across segments.

What you do

  • Segment explicitly (preferred in production).
  • Or fit mixture (Gaussian mixture, etc.) for analysis.
  • Monitor per-segment; global metrics can hide failures.

Heuristic

  • If “overall average” doesn’t describe any real user group, suspect a mixture.

Categorical / Multinomial

Categorical

  • Single draw from K categories.

Multinomial

  • Counts over K categories across N draws.

Where it appears

  • Label distribution, token frequencies, error codes, country/device distributions.
  • Drift detection: “did category mix change?”

Recognize

  • Finite set of labels/categories; you track proportions.

What you do

  • Chi-square tests / G-tests; JS divergence; Dirichlet smoothing.
  • Watch for rare categories (small expected counts → Fisher/Monte Carlo).

Heuristic

  • For categorical drift, start with simple proportion monitoring + chi-square/JS; don’t overcomplicate.

A few cross-cutting heuristics (high leverage)

  1. Check support first: is it {0,1}, nonnegative integers, positive reals, [0,1], simplex? That narrows choices fast.

  2. Mean vs variance relationship:

    • Poisson: var ≈ mean
    • NB: var > mean

  3. Tail behavior decides your stats:

    • Heavy tails → quantiles/robust/bootstrap

  4. Heterogeneity beats “fancy fitting”:

    • If mixture suspected, segment first.