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 ...