(independence of events in disjoint intervals for Poisson law) The average number of cars arriving at a tollbooth per minute is λ and the probability of k cars in the interval (0, T) minutes is Consider two disjoint, that is, nonoverlapping, intervals, say (0, 1] and ( , T]. Then for the Poisson law: P[ cars in (0, ] and cars in ( , T]]= P[ cars in (0, ]]P[ cars in ( , T]], that is events in disjoint intervals are independent. Using this fact, show the following: (a) That P[n1 cars in (0, t1]|n1 + n2 cars in (0, T]] is not a function of λ. (b) In (a) let T = 2, t1 = 1, n1 = 5, and n2 = 5. Compute P[5 cars in (0, 1]|10 cars in (0, 2].