Problem 20
Consider a Poisson process with intensity \(\lambda\), and let \(T\) be the time of the last occurrence in the time interval \((0, t]\). If there is no occurrence during \((0, t]\), we set \(T=0\). Compute \(E T\).
Problem 21
A further (and final) definition of the Poisson process runs as follows: A nondecreasing stochastic process, \(\\{X(t), t \geq 0\\}\), is a Poisson process iff (a) it is nonnegative, integer-valued, and \(X(0)=0\); (b) it has independent, stationary increments; and (c) it increases by jumps of unit magnitude only. Show that a process satisfying these conditions is a Poisson process.
