(Another direction from Exercise 1.13) a) For any given n ≥ 2, find the probability that X n and X n+1 are both records-to-date. Hint: The idea in part b) of 1.13 is helpful here, but the result is...

a) For any given n ≥ 2, find the probability that X_n
and X_n+1
are both records-to-date. Hint: The idea in part b) of 1.13 is helpful here, but the result is not.

b) Is the event that X_n
is a record-to-date statistically independent of the event that X_n+1
is a record-to-date?

c) Find the expected number of adjacent pairs of records-to-date over the sequence X₁, X₂, . . . .

Exercise 1.13

Let X₁, X₂, . . . , X_n, . . . be a sequence of IID continuous rv’s with the