Question

8.186 The number N(t) of phone calls arriving at a switchboard during the first t minutes time that elapses between when you start your stopwatch and when the nth phone call arrives. after you start your stopwatch has a Poisson distribution with parameter 3.8t. Let W be the a) On average, how many phone calls arrive during the first t minutes? b) If it is known that Wi > t, what can be said about N(t)? Similarly, what would Wit imply about N(t) ? c) Use part (b) and your knowledge of the Poisson distribution to find P(W > t). d) Use part (c) to identify the probability distribution of W1. e) Determine and identify the probability distribution of W3. Hint: See the note directly preceding Example 8.16 on page 454. ) Determine and identify the probability distribution of Wh. 8.187 To obtain the area of a circle, its diamate mant

Answer 1

(a) Observe that N(t) ~Poisson(3.8t). Then the average number of calls arived during first t minutes is E[N(t)] = 3.8t (b) If W> t, then it means that no calls has appeared in time t. Then N(t) 0. Similarly W <t N(t)1 From part b) (c) get we P(W1 t) P(N (t) 0) 3.8t = e From part c) (d) we get Fwi (t)1 P(W1 t) =1 -e 3.8t Then a pdf of Wi would be d (Fw,(t) dt fwi(t) 3.8e 3.8t Hence, W Exp(3.8)