11 Consider a Yule process starting with a single individual—that is, suppose X(0) = 1. Let Tt denote the time it takes the process to go from a population of size / to one of size / + 1.

(a) Argue that Ti9 / = l , . . . , y , are independent exponentials with respective rates /A.

(b) Let Xx,..., Xj denote independent exponential random variables each having rate A, and interpret Xt as the lifetime of component /. Argue that max (Xx,..., Xj) can be expressed as m a x t A ^ , X j ) = å÷ + å2 + ··· + â,·