The property of independence and identical distribution ("IID") is a fundamental concept in probability and statistics. It describes a collection of [[Random Variable|random variables]] with two key properties: **Independence:** A set of r.v's. $X_1, \dots, X_n$ are [[Independence of Random Variables|independent]] if the outcome of any $X_i$ does not influence the outcome of any other $X_j$ for $i \neq j$. Mathematically, this is expressed as: $ \mathbf P(X_1=x_1, \dots, X_n=x_n)= \prod_{i=1}^n \mathbf P(X_i=x_i)$ **Identical Distribution:** A set of r.v's. $X_1, \dots, X_n$ are identically distributed if they come from the same population distribution. This means all $X_i$ have the same distribution parameters (e.g. mean $\mu$ and variance $\sigma^2$).