Conditional Distributions in the GLM

The core principle of the GLM: for every x-value a distinct conditional distribution P(Y|x) arises. The linear predictor η = β₀ + β₁·x is transformed via the link function into the distribution parameter — which then determines the shape and location of the distribution. η = β₀ + β₁·x can take any value. The link function transforms it into the valid parameter range:

Log link: λ = e^η — always positive (Poisson, Gamma)
Logit link: p = 1/(1+e^−η) — always in (0,1) (Bernoulli, Binomial)
Identity: μ = η — the classical LM (Normal)

• Single: conditional distribution for the selected x-value
• Ghost: all x-values overlaid — shape change visible
• Ridgeline: all distributions stacked — full picture at a glance

Normal (OLS): continuous data, unbounded · σ constant
Poisson / Neg. Binomial: count data (0,1,2,…); NB for overdispersion
Bernoulli / Binomial: 0/1-data or k successes from n trials
Gamma: positive continuous data, right-skewed
ZIP / Hurdle-Poisson: count data with many zeros — two processes combined Reflection questions appear at the bottom to deepen understanding — e.g. why variance and mean are identical for Poisson or when to choose Neg. Binomial instead of Poisson.