Let us revisit the record value problem of Section 3.2.3. Make sure that all reasonings there remain true in the case where X’s are continuous. Show that in this case

and that P(N

for n = 1,2, ... (Advice: Show that, since X’s are independent and continuous, with probability one, among the r.v.’s X₁, ...,X_n, only one r.v. attains the largest value.)

We understand that we consider the r.v. N for the case when X₀ is random; that is, the value of X₀ is a future value for us, as well as values of other X’s. What is the conditional distribution of N given that X₀ has assumed a particular value x₀?