Exchangeability vs Independent and Identically Distributed (IID)

Exchangeability Does NOT Imply IID

The following is an exchangeable but not IID.

Consider an urn with a red ball and a blue ball; with a probability 1/2 of drawing either. Balls are drawn without replacement. Let:

• $Y_i=\{\begin{array}{ll}1& \text{if the }i\text{th ball is red}\\ 0& \text{otherwise}\end{array}$

Since 𝐏(𝑦₁=1,𝑦₂=0) = 𝐏(𝑦₁=0,𝑦₂=1) then {𝑦₁,𝑦₂} are exchangeable.

However, they are not independent because:

• 𝐏(𝑦₂=1|𝑦₁=1) ≠ 𝐏(𝑦₂=1)
• 0 ≠ 1/2

IID Implies Exchangeability

If {𝑦₁, 𝑦₂, …, 𝑦_𝑛} are IID, then they are exchangeable.