The distribution of a [[Random Variable]] $Y$ is a member of the exponential family, when its [[Probability Density Function|PDF]] $f_\theta(y)$ can be expressed in the following form. $ \begin{align} f_\theta (y) &= h(y) \exp \Big( T(y)\eta (\theta)- B(\theta) \Big) && (\mathrm{if}\,\theta \in \mathbb R)\\[6pt] f_\theta (y) &= h(y) \exp \left [\sum_{i=1}^k ( T_i(y)\eta_i (\theta)- B(\theta) \right] && (\mathrm{if}\, \theta \in \mathbb R^k) \end{align} $ The components of this representation are: - $h(y)$: A scaling function that only depends on the data $y$. - $\exp(\cdot)$: An exponential function that is govern by two elements. - $T(y)\eta(\theta)$: An interaction of a function depending on the data $T(y)$ with a function depending on the distribution parameter $\eta(\theta)$. - $B(\theta)$: A function depending on the distribution parameter $\theta$. **Bernoulli:** $ \begin{align} f_p(y) &= p^y*(1-p)^{1-y}\\[6pt] &=\exp \Big\{y \ln(p)+(1-y) \ln(1-p)\Big\} \\[6pt] &=\exp \Big\{y \ln(p)+\ln(1-p)-y\ln(1-p)\Big \} \\[6pt] &=\exp \Big\{y*\ln(\frac{p}{1-p})+\ln(1-p)\Big \} \\[6pt] &=\exp \Big\{\underbrace{y}_{ T(y)}*\underbrace{\ln(\frac{p}{1-p})}_{\eta(p)}-\underbrace{(-\ln(1-p))}_{B(p)}\Big \}*\underbrace 1_{h(y)} \\[6pt] \end{align} $ **Poisson:** $ \begin{align} f_\lambda(y)&=\frac{\lambda^y}{y!}*e^{-\lambda} \\[6pt] &=\underbrace{\frac{1}{y!}}_{h(y)}*\exp\Big\{\underbrace{y}_{T(y)}* \underbrace{\ln(\lambda)}_{\eta(\lambda)}- \underbrace{\lambda}_{B(\lambda)} \Big\} \end{align} $ **Binomial:** (assuming fixed $n$) $ \begin{aligned} f_p(y) &= \binom{n}{y} p^y (1-p)^{n-y} \\[8pt] &=\binom{n}{y}\exp\Big\{y \ln(p)+(n-y) \ln(1-p) \Big\}\\[8pt] &=\binom{n}{y}\exp\Big\{y \ln(p)+n\ln(1-p)-y\ln(1-p) \Big\}\\[8pt] &=\binom{n}{y}\exp\Big\{y \ln(\frac{p}{1-p})+n\ln(1-p)\Big\}\\[8pt] &=\underbrace{\binom{n}{y}}_{h(y)}\exp\Big\{\underbrace{y}_{T(y)}* \underbrace{\ln(\frac{p}{1-p})}_{\eta(p)}-\underbrace{(-n\ln(1-p))}_{B(p)}\Big\}\\[8pt] \end{aligned} $