2.2 Standard Random Sample

Definition 2.1 (Standard Random Sample) The random variables \(X_i\), \(i = 1,\dots , n\) are called standard random sample of size \(n\) from population \(f(x)\) if \(X_i\)’s are I.I.D. RVs from the same probability density function \(f\).

There is a few nuisances regarding general practice in statistics and this definition.

1. Definition 2.1 is either for infinite population or finite population with sampling with replacement.

2. For finite population of size \(n\), sample data \(X_i\) from sampling without replacement can never be independent as \(\mathbb{P}(X_2 = y | X_1 = y) = 0\) and \(\mathbb{P}(X_2 = y | X_1 = x) = 1/(n-1)\).

In this course, when we talk about sampling, we will understand it as in Definition 2.1.