Let X1,X2, . . . be independent and identically distributed nonnegative continuous random variables having density function f (x). We say that a record occurs at time n if Xn is larger than each of the previous values X1, . . . , Xn−1.

(A record automatically occurs at time 1.) If a record occurs at time n, then Xn is called a record value. In other words, a record occurs whenever a new high is reached, and that new high is called the record value. Let N(t) denote the number of record values that are less than or equal to t . Characterize the process {N(t), t ≥ 0} when

(a) f is an arbitrary continuous density function.

(b) f (x) = λe

−λx.

Hint: Finish the following sentence: There will be a record whose value is between t and t + dt if the first Xi that is greater than t lies between . . .