三点独立同分布：上涨后更可能下跌

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 $\mathbf{B} > \mathbf{A}$ , and a decrement otherwise. Similarly, an increment occurs with C if $\mathbf{C} > \mathbf{B}$ , 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 ex- changeable, then their

density, say f(a, b, c), is the same for any permutationof (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 $\mu$ and variance, and let them have the common pairwise correlation $p$ . Show that the correlation of the rvs B-A and C-B is generally equal to $-\frac{1}{2}$ .