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Module 10/Joint distributions

Exchangeable random variables

A sequence is exchangeable when reordering it doesn't change its distribution. That symmetry lets you compute positional probabilities — like ‘‘the 23rd card is a spade’’ — as if they were the first.

Before you start — give these a try

Attempting first primes your brain for the lesson — even if you miss. Nothing is graded or saved; it's just a warm-up.

A well-shuffled 5252-card deck is dealt one card at a time. Find P(the 23rd card is a spade)P(\text{the 23rd card is a spade}).

Any i.i.d. sequence of random variables is exchangeable.

What you’ll be able to do

  • Define exchangeability via equality in distribution under permutations.
  • Use the symmetric-joint-p.m.f. test (Proposition 15.5).
  • Exploit exchangeability to reduce positional probabilities to the first few draws.

In your course

· MATH2015 · Linear Algebra & Probability
§15.3 Exchangeable Random Variables
  • Definition 15.3Exchangeable random variables
  • Proposition 15.5Exchangeable ⟺ symmetric joint p.m.f.
  • Proposition 15.6i.i.d. sequences are exchangeable
  • Example 15.4 / 15.623rd card is a spade (1/41/4); urn draws
1

Exchangeability

X1,…,XnX_1,\dots,X_n are exchangeable if for every permutation (k1,…,kn)(k_1,\dots,k_n) of (1,…,n)(1,\dots,n),\n\n(X1,…,Xn)=d(Xk1,…,Xkn),(X_1,\dots,X_n)\stackrel{d}{=}(X_{k_1},\dots,X_{k_n}),\n\ni.e. reordering leaves the joint distribution unchanged. A consequence: all the XjX_j share the same marginal distribution.

2

How to check it

X1,…,XnX_1,\dots,X_n are exchangeable iff their joint p.m.f. is a symmetric function (Proposition 15.5): p(xk1,…,xkn)=p(x1,…,xn)p(x_{k_1},\dots,x_{k_n})=p(x_1,\dots,x_n) for every permutation. Two big families are exchangeable: any i.i.d. sequence (Proposition 15.6), and labels drawn without replacement from an urn.

3

Why it helps

Under exchangeability you may rearrange variables freely inside a probability or expectation. So the chance the 23rd card is a spade equals the chance the 1st card is — namely 1/41/4 — and positional draws reduce to the first few positions.

Proposition 15.5 — Checking exchangeability

X1,…,XnX_1,\dots,X_n (discrete, joint p.m.f. pp) are exchangeable iff pp is symmetric under every permutation of its arguments.

Intuition. Symmetry of the joint law is exactly invariance under reordering.
Proposition 15.6 — i.i.d. ⟹ exchangeable

Any i.i.d. sequence of random variables is exchangeable.

Intuition. Identical, independent pieces look the same in any order.

Worked examples

Example 1

A shuffled deck is turned over one card at a time. What is P(23rd card is a spade)P(\text{23rd card is a spade})?

  1. 1

    The card sequence is exchangeable, so position 2323 has the same marginal as position 11.

  2. 2

    P(1st card is a spade)=13/52=1/4P(\text{1st card is a spade})=13/52=1/4.

  3. 3

    Hence P(23rd is a spade)=1/4P(\text{23rd is a spade})=1/4.

Answer. 1/4=0.251/4=0.25.
Example 2

Urn with 3737 red, 6161 green, 5050 yellow (148148 total). Draw without replacement. Find P(9th yellow,12th red,20th yellow)P(9\text{th yellow},12\text{th red},20\text{th yellow}).