In the field of probability and statistics, it is not uncommon to encounter situations where the probability distribution function cannot be calculated directly. One example of such a situation is the Bayesian inference.
A popular example of Bayesian inference is the estimation of model parameters. The parameters are the elements that define the model structure. For example, for a normal distribution, the parameters are the mean and the variance. For an exponential distribution, the parameter is λ.
In this case, we usually have some samples x that come from a distribution with unknown parameters θ, but we assume we have some prior knowledge about θ we can express using a probability function p(θ). Using Bayes theorem, we can find the probability for θ according to observed data x.