The Problem Traditionally, the value of a position in a chess engine is computed as the maximum of the values of the subsequent positions. These values are typically represented as a single number. However, I am exploring an alternative approach where a position's value is characterized as a distribution. The goal is to find a distribution that maintains remains the same type even after taking the maximum.
Question: Let's say we have a family of distributions $H$. Consider $X_1,X_2,...X_n$ as independent random variables, not necessarily identically distributed, with distributions from $H$. Is it possible for the maximum of these variables, $Max(X_1,X_2,...X_n)$, to also have a distribution from $H$, where H is not a trivial solution (e.g. the set of all distributions)?
The family of Inverse Exponential Distributions has this property. Let $X_1, X_2, \dots, X_n$ be independent random variables such that $X_k$, $k\in\{1,2,\dots,n\}$, follows an inverse exponential distribution of parameter $\lambda_k > 0$.
The pdf and cdf of $X_k$ is given by $f_{X_k}(x) = \frac{\lambda_k}{x^2} e^{-\lambda_k/x}$ and $F_{X_k}(x) = e^{-\lambda_k/x}$, respectively.
Let $Y = \max\{X_1,X_2,\dots,X_n\}$. Then, the cdf of $Y$ is given by \begin{align} F_Y(y) = \prod_{k=1}^n F_{X_k}(y) = \exp\!\left(-\sum_{k=1}^n \frac{\lambda_k}{x}\right) = \exp\!\left(-\frac{\sum_{k=1}^n \lambda_k}{x}\right). \end{align}
Thus, $Y$ follows an inverse exponential distribution of parameter $\lambda = \sum_{k=1}^n \lambda_k$.