*11. Consider a Yule process starting with a single individual—that is,

supposeI mage. Let denote the time it takes the process to go from a population of size i to one of size .

(a) Argue that Image, are independent exponentials with respective rates iλ.

(b) Let Image denote independent exponential random variables each having rate λ, and interpret as the lifetime of component i. Argue that Image can be expressed as Image where are independent exponentials with respective rates jλ, Image.

Hint: InterpretI mage as the time between the and the ith failure.

(c) Using

(a) and

(b) argue that Image

(d) Use

(c) to obtain and hence, given Image has a geometric distribution with parameter Image.

(e) Now conclude that