5.9 Gamma-Poisson conjugate family 6/8

5.9.1 Gamma prior : Gamma and Exponential models

$\tau$ continuous random variable but can only take + value

$$\tau \sim Gamma(s, r)$$

Probability density functions:

$$f(\tau) = \frac{r^s}{\Gamma (s)} \tau^{s - 1} e^{-r\tau} \quad for \quad \tau > 0$$ $$E(\tau) = \frac{s}{r} ; Mode(\tau) = \frac{s - 1}{r} \quad for \quad s \geq 1; Var(\tau) = \frac{s}{r^2}$$

When s = 1 -> Exponential model = Gamma(1,r)

$$\tau \sim Exp(r)$$