Ups and downs occur in probability just as in real life. An elementary probabili

Question

Ups and downs occur in probability just as in real life. An elementary probability
version with the real-life property that ups are (more) often followed
by downs is as follows.
Let (A,B,C) be three independent and identically distributed continuous
rvs that are realized sequentially: first A, then B, and finally C. Let us say that
an increment occurs with B if B > A, and a decrement otherwise. Similarly,
an increment occurs with C if C > B, and a decrement otherwise.
a. Suppose you are told only that B has led to an increment, but not the
actual value of B. Argue that conditional on this information C is twice as
likely to yield a decrement than an increment — even though B and C are
independent.
b. Random variables are said to be exchangeable if their joint density is
the same for any permutation of its arguments. Thus, if (A,B,C) are exchangeable,
then their density, say f(a, b, c), is the same for any permutation
of (a, b, c). Note that exchangeable rvs may still be dependent. Does the property
described in a. still hold if (A,B,C) are exchangeable?
c. Suppose that the three rvs have the same marginal distribution with
mean and variance 2, and let them have the common pairwise correlation
. Show that the correlation of the rvs B A and C B is generally equal
to 1/2 .

Explanation / Answer

a. Consider the six possible rank orders of (A,B,C); each of these ordersis equally likely because the rvs were assumed to be identically and independentlydistributed. If, e.g., abc stands for the increasing order A < B < C,then three of the six orders are such that A < B, so that an increment occurswith B. These are those orders in which the letter a occurs before b, namely,abc, acb, cab. The mere fact that there are three (an odd number) such ordersalready implies that following an increment with B there cannot be an equalprobability for increments and decrements with C. Indeed, only the case abcyields a further increment, whereas acb and cab represent a decrement, i.e.,C < B. Therefore, the conditional probability of a decrement with C, givenan increment at B is 23 : despite the independence of all rvs involved, ups aremore often followed by downs than by a further up.

b. Let f(a, b, c) denote the joint density of (A,B,C). The assumptionsstated in question b. imply that, e.g., f(a, b, c) = f(c, a, b), and similarly forany permutation of the three arguments. Therefore, each of the six possiblerank orders must still be equally likely, so that the argument from a. still goesthrough, even for dependent but exchangeable rvs.

c. To find this correlation, we first determine the associated covarianceCov(B A,C B) = Cov(B,C) Var(B) Cov(A,C) + Cov(A,B)= 2 2where 2 denotes the (common) covariance between any pair of the three rvs.Also, from standard linearity properties of the variance we find

Var(B A) = Var(B) + Var(A) 2 Cov(B,A)= 2(2 2)and for reasons of symmetry, Var(CB) = 2(2 2). Therefore, the correlation

corr(B A,C B) = Cov(B A,C B)
Var(B A) · Var(C B)

=-1/2

The successive differences (B A) and (C B) are negatively correlatedbecause B occurs in both expressions, but with opposite signs. For example,if B is particularly large, then (B A) will usually be positive and (C B)will usually be negative.

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