Log-probability derivation of max/min of non-i.i.d. random variables

[larryshamalama]

Description

This builds upon work on automatic log-probability derivation of max/min statistics. We would like to extend this to obtaining the maximum/minimum order statistics of non-i.i.d. (univariate) random variables.

We hope to deconstruct this issue into the following incremental PRs:

• Maximum/minimum of two random variables from the same distribution family, e.g. $X \sim \text{Normal}(\mu_X, \sigma_X^2); Y \sim \text{Normal}(\mu_Y, \sigma_Y^2)$.
• Maximum/minimum of two random variables from different distribution families, but with the same support, e.g. $X \sim \text{Normal}(\mu, \sigma^2); Y \sim \text{StudentT}(\nu)$.
• Maximum/minimum of two random variables from different distribution families with possibly different support, e.g. $X \sim \text{Normal}, Y \sim \text{Gamma}$ or even $X \sim \text{Normal}, Y \sim \text{Bernouilli}$.
• Maximum/minimum of $n$ random variables from the same distribution family.
• Maximum/minimum of $n$ random variables from different distribution families, with the same support.
• Maximum/minimum of any $n$ univariate random variables.

As we progress, each task can possibly deserve their own issue to keep track of mathematical derivations and code templates. For prospective GSoC applicants, the realm of log-probability for order statistics (in PyMC) is vast and the bullet point list above is ambitious.

CC @ricardoV94 @Dhruvanshu-Joshi