Notation. Finite sets, maps, distributions, random variables, and expectation [ftip-000A]
Notation. Finite sets, maps, distributions, random variables, and expectation [ftip-000A]
For a finite set \(X\), write \(|X|\) for its cardinality and \(X^*\) for the set of finite sequences with entries in \(X\). A map \(f:X\to Y\) sends \(x\in X\) to \(f(x)\in Y\); the inverse image of \(A\subseteq Y\) is \(f^{-1}(A)=\{x\in X:f(x)\in A\}\).
Write \[ \Delta (X)=\left \{p:X\to [0,1]:\sum _{x\in X}p(x)=1\right \} \] for the probability simplex on \(X\). If \(p\in \Delta (X)\) and \(Z:X\to \mathbb R\), then \[ \mathbb E_{x\sim p}[Z(x)]=\sum _{x\in X}p(x)Z(x). \] A random variable is a map from the sample set to its value set. Random variables \(Z\) and \(W\) are independent under \(p\) when \(p(Z=z,W=w)=p(Z=z)p(W=w)\) for every pair of values \(z,w\).
For nonfinite interaction spaces, probability laws and conditional kernels require the measurable structure described in Remark [ftip-000C].