IID Random Variables

Compact study note.

Summary

IID means random variables are independent and share the same distribution. This assumption powers sample means, laws of large numbers, and the central limit theorem.^[1]

Prerequisites

• Independence
• Random Variable

Notation and Assumptions

$X_1,\dots ,X_n$ are IID when they are mutually independent and each $X_i$ has the same distribution.

Essential Result

For IID variables with finite mean $\mu$ and variance ${\sigma }^2$ , $E[{\overline{X}}_n]=\mu$ and $\mathrm{Var}({\overline{X}}_n)={\sigma }^2/n$ .

Small Example

Repeated fair die rolls are IID if the die is rolled under identical conditions and previous rolls do not influence later rolls.

Common Mistakes

• Checking identical distribution but forgetting independence.
• Assuming time-ordered data are IID without checking dependence or drift.

Connections

• Law of Large Numbers
• Central Limit Theorem
• Binomial Distribution

References