Let X1, . . . , be independent and identically distributed continuous positive random variables. Suppose these random variables are observed in sequence, and say that Xj is an upper record value if Xj = max(X1, . . . , Xj ). That is, an upper record value is one that is larger than all previous values. Let Rn be the nth upper record value when the Xi are all exponential with rate λ.

(a) What is the distribution of R2 − R1, the amount by which the second record value exceeds the first?

(b) What is the distribution of Rn?