Section 7.2

Understand the definition of Exchangeable Random Variables.

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Section 7.2: Exchangeable Random Variables

First, recall Definition 5.12 for equality in distribution of two real-valued random variables. Analogously, we say that two random vectors $LaTeX: (X_1, \ldots, X_n)$ and $LaTeX: (Y_1, \ldots, Y_n)$ are equal in distribution if

$LaTeX: P((X_1,\ldots, X_n)\in B)=P((Y_1, \ldots, Y_n) \in B)$

for all subsets B of $LaTeX: \mathbb{R}^n$ . We abbreviate this as

$LaTeX: E\left[g\left(X_1, \ldots, X_n\right)\right]=E\left[g\left(Y_1, \ldots, Y_n\right)\right]$ ,

$LaTeX: (X_1,\ldots,X_n) =^d (Y_1,\ldots, Y_n)$

Definition 7.15. A sequence of random variables $LaTeX: X_1, X_2, \ldots, X_n$ is exchangeable if the following conditions holds: for every permutation ( $LaTeX: (k_1, k_2, \ldots, k_n)$ of ( $LaTeX: (1,2,\ldots,n)$ , we have

$LaTeX: (X_1, X_2,\ldots,X_n) =^d (X_{k1},X_{k2},\ldots, X_{kn})$

Example 7.16. We illustrate Definition 7.15 for n = 3. The 6 permutations of (1,2,3) are

(1,2,3), (1,3,2), (2,1,3), (2,3,1) (3,1,2), (3,2,1).

Thus the random variables LaTeX: X_1,X_2,X_3 are exchangeable if and only if the following six vectors all have the same distribution:

$LaTeX: \begin{array}{lll} \left(X_1, X_2, X_3\right), & \left(X_1, X_3, X_2\right), & \left(X_2, X_1, X_3\right) \\ \left(X_2, X_3, X_1\right), & \left(X_3, X_1, X_2\right), & \left(X_3, X_2, X_1\right) \end{array}$

Having stated this somewhat abstract definition of exchangeability, we address two questions.

How do we check for exchangeability?
What does exchangeability allow us to do in calculations?

Fact 7.17 (How to check exchangeability)

Suppose $LaTeX: X_1, X_2, \ldots, X_n$ are discrete random variables with joint probability mass function p. Then these random variables are exchangeable if and only if p is a symmetric function.

Suppose $LaTeX: X_1, X_2, \ldots, X_n$ are jointly continuous random variables with joint density function f. Then these random variables are exchangeable if and only if f is a symmetric function.

In mathematics, a function of n variables is symmetric if its value is the same no matter the order of its arguments.

Fact 7.18. Suppose that $LaTeX: X_1, X_2, \ldots, X_n$ are exchangeable. Then the following statements hold.

The random variables are identically distributed, that is, they have the same marginal distributions. More concretely, if the random variables are discrete then each has the same marginal probability mass function, while if they are continuous then each has the same marginal density function.
For any $LaTeX: 1\le k \le n$ the random variables $LaTeX: (X_1, \ldots, X_k)$ have the same joint distribution as any other k-tuple $LaTeX: (X_{i_1}, X_{i_2}, \ldots, X_{i_k})$ . Consequently, for any function g of k variables, the expectation does not change under permutations: $LaTeX: E[g(X_1, X_2, \ldots, X_k)] = E[g(X_{i_1}, X_{i_2}, \ldots, X{i_k})]$

Independent identically distributed random variables

We begin with the basic definition.

Definition 7.19. Random variables $LaTeX: X_1, X_2, X_3, \ldots$ are independent and identically distributed (abbreviated i.i.d.) if they are independent and each LaTeX: X_k has the same probability distribution.

Theorem 7.20. An i.i.d. sequence of random variables is exchangeable.

Example 7.21. Suppose that LaTeX: X_1, X_2, X_3 are independent Unif[0,1] random variables. Find the probability that is the largest.

Example 7.21. Suppose that LaTeX: X_1, X_2, X_3 are independent Unif[0,1] random variables. Find the probability that is the largest.

Sampling without replacement

Theorem 7.22. Let $LaTeX: 1\le k \le n$ . Let $LaTeX: X_1, \ldots, X_k$ are exchangeable. Then, for any function g, the sequence of random variables $LaTeX: g(X_1), \ldots, g(X_n)$ is exchangeable.

Notes

By using exchangeability, compute the probabilities.

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Section 7.2: Exchangeable Random Variables

Example 7.24. Suppose an urn contains 37 red balls, 61 green balls, and 50 yellow balls. We sample 20 balls without replacement. What is the probability that the 9th draw is yellow, the 12th draw is red, and the 20th draw is yellow?

Example 7.25. Suppose we deal 10 cards from a standard deck of 52. What is the probability that the sixth card is an ace, given that the fifth and tenth cards are both aces.

Notes