Exponentially distributed lifetimes have constant hazard rate equal to the rate parameter \lambda . When \lambda is a constant hazard rate, a simple way to model heterogeneity of hazards is to introduce a multiplicative frailty parameter \mu , so that lifetimes T_i have distribution Ti~ f(til, u) = \ exp{-Aut;}, t; > 0, 1,4 > 0 .

The prior on (\lambda, \mu) is \pi(\lambda, \mu) \propto \lambda^{c-1} \mu^{d-1} \cdot \text{exp}\{-\alpha \lambda - \beta \mu\} , that is, \lambda and \mu are a priori independent with distributions \mathcal{G}\text{a}(c, \alpha) and \mathcal{G}\text{a}(d, \beta ) , respectively. The hyperparameters c, d, \alpha, and \beta are known (elicited) and positive.

Assume that lifetimes t_1, t_2,..., t_n are observed. Show that full conditionals for \lambda and \mu are gamma, [\lambda | \mu, t_1, ... , t_n] \sim \mathcal{G}\text{a} \left(n+c, \mu \sum_{i=1}^{n} {t_i} + \alpha \right), and by symmetry, [\mu | \lambda, t_1, ... , t_n] \sim \mathcal{G}\text{a} \left(n+d, \lambda \sum_{i=1}^{n} {t_i} + \beta \right).