Poisson Processes 241 5.2.11 Let X and Y be jointly distributed random variables and B an arbitrary set. Fill in the details that justify the inequality | Pr{X in B} - Pr{Y in B}| Pr{X = Y}. Hint: Begin with {X in B} = {X in B and Y in B} or {X in B and Y not in B} {Y in B} or {X = Y}. 5.2.12 Computer Challenge Most computers have available a routine for simulat- ing a sequence U 0 , U 1 , . . . of independent random variables, each uniformly distributed on the interval (0, 1). Plot, say, 10,000 pairs (U 2n , U 2n+1 ) on the unit square. Does the plot look like what you would expect? Repeat the exper- iment several times. Do the points in a fixed number of disjoint squares of area 1/10,000 look like independent unit Poisson random variables? 5.3 Distributions Associated with the Poisson Process A Poisson point process N((s, t]) counts the number of events occurring in an interval (s, t]. A Poisson counting process, or more simply a Poisson process X(t), counts the number of events occurring up to time t. Formally, X(t) = N((0, t]).