What are the traditional distributions for assigning probabilities to model parameters? For instance, assume that we have a binomial distribution: $$y \sim Bin(n,\theta)$$ Then we can distribute $\theta$ parameter with a Beta distribution and parameters $\alpha$ and $\beta$ equal to 1. $$\theta \sim Beta(1,1)$$ I consider that if we would like to derive the distribution of the mean of a Gaussian probability distribution, then we could use the central limit theorem and assume that: $$\mu \sim N(0,1)$$ But what about other distributions? For instance, what is the distribution of $\lambda$ in the Weibull distribution? Or what is the distribution of $k$ for Chi-squared distribution? I'm not sure where to find papers on this topic because in most cases the usage of techniques involving parameters distributions seems like more of an intuitive methodology than some kind of rule.
Update: I'm asking this question for the Bayesian parameters estimation and Bayesian model selection, where I calculate posterior with: $$P(\theta|y) = P(y|\theta)P(\theta)$$ So I need to find the prior $P(\theta)$, that's what I was interested in.