March 10, 2017 Math 303 Assignment 8: Due Friday, March 17 at start of class Test 2 will be held in class on Wednesday March 22, and will be based on the material covered in Assignments 5–8. No assignment is due March 24. Assignment 9, which is the last one, will be available March 24. I. Problems to be handed in: 1. Let {N (t) : t ≥ 0} be a Poisson process of rate λ. For 0 ≤ u ≤ t and n = 1, 2, . . ., determine the conditional probability of N (u) given that N (t) = n. 2. Let {N (t) : t ≥ 0} be a Poisson process of rate λ. Given that N (t) = 3, determine the conditional (marginal) probability density functions of each of the three arrival times S1 , S2 , S3 . 3. Customers arrive at a facility according to a Poisson process {N (t) : t ≥ 0} of rate λ, with arrival times S1 , S2 , . . .. Each customer pays $1 on arrival. At time t, the discounted value of the total PN (t) sum collected so far, discounted back to time zero, is i=1 e−βSi , where β > 0 is the discount rate. Show that the discounted value at time t has expectation λβ −1 (1 − e−βt ). 4. Alpha particles are emitted from a radioactive source according to a Poisson process of rate λ. Each alpha particle independently exists for a random duration and then is annihilated. The lifetimes Y1 , Y2 , . . . of the particles have common cumulative distribution function G(y) = P (Yk ≤ y). Let M (t) denote the number of alpha particles in existence (not yet annihilated) at time t. (a) Determine the distribution of M (t). (b) Show that in the long run (i.e., in the limit t → ∞) the distribution of part (a) is Poisson(λµ) where µ is the mean lifetime of the alpha particles. 5. (a) Given that a Poisson process of rate λ has exactly n events by time T , show that the conditional distribution of the time of the nth event has density T −n nsn−1 (0 ≤ sn ≤ T ). n (b) We wish to estimate the rate λ in a Poisson process {N (t) : t ≥ 0}, as follows. First, we fix some time threshold T > 0, and then conditional on N (T ) > 0, we observe the interarrival 1 1 times Xi and set X̄N (T ) = N (T ) (X1 + · · · + XN (T ) ). We use X̄N (T ) as our estimate for λ . Show that this is a biased estimator, by showing that 1 λT 1 E[X̄N (T ) | N (T ) > 0] = 1 − λT 6= . λ e −1 λ II. Recommended problems: These provide additional practice but are not to be handed in. Chapter 5 #60*, 64* Quote of the week: Since you are now studying geometry and trigonometry, I will give you a problem. A ship sails the ocean. It left Boston with a cargo of wool. It grosses 200 tons. It is bound for Le Havre. The mainmast is broken, the cabin boy is on deck, there are 12 passengers aboard, the wind is blowing East-North-East, the clock points to a quarter past three in the afternoon. It is the month of May. How old is the captain? Gustave Flaubert