Stat 432: Homework 8 1. Below is drawing of the intersection of Main St., North St. and South St. Main St. North St. South St. Cars driving on Main St come to a T-intersection and have to choose either North St. or South St. to continue. Assume that the cars on Main St. arrive at the T-intersection according to a Poisson process with an average rate of 1 every 4 minutes. A car on Main St. arriving at the T-intersection chooses North St. with probability 2/3 and South St. with probability 1/3. Let tNM count the number of cars on Main St. arriving at the T-intersection. Let tNMN count the number of cars on Main St. choosing North St. to continue. Let tNMS count the number of cars on Main St. choosing South St. to continue. a) What is the probability that 10 cars on Main St. arrive at the T-intersection in a given one-hour period? b) What is the probability that 10 cars on Main St. choose North St. to continue in a given one-hour period? c) What is the probability that 10 cars on Main St. choose South St. to continue in a given one-hour period? d) What is the probability that the time to the first arrival at the T-intersection of a car on Main St. is less than 10 minutes? e) What is the probability that the time to the first car on Main St. to choose North St. is less than 10 minutes? f) What is the probability that the time to the first car on Main St. to choose South St. is less than 10 minutes? g) What is the probability that the time to the second car on Main St. to choose North St is less than 10 minutes? 2. Cars on North St. (travelling south) arrive at the intersection of Main St. according to a Poisson process with an average rate of 1 every 3 minutes. Cars on South St. (travelling north) arrive at the intersection with Main St. according to a Poisson process with an average rate of 1 every 6 minutes. Assume that arrivals at the intersection from the three directions are independent. 12a) What is the probability that less than 40 cars arrive at the intersection in a given one-hour period? b) What is the probability that the time to the first arrival of a car at the intersection is more than 2 minutes? c) What is the probability that the time to the second car to arrive at the intersection is less than 2 minutes? d) What is the probability that 2 cars arrive at the intersection from the north before a car arrives at the intersection from the south? 3. In problem 1 of homework 6, we looked at a model for social mobility. The one step transition probabilities for the three states; Professional (P), Skilled (S) and Unskilled (U) are given in the matrix below. P S U P 0.80 0.15 0.05 S 0.25 0.60 0.15 U 0.20 0.30 0.50 Assume that in a small community, initially, the number of Professionals is a Poisson random variable with mean 12, the number of Skilled workers is a Poisson random variable with mean 36, and the number of Unskilled workers is a Poisson random variable with mean 52. These Poisson random variables are assumed to be independent. a) Describe the distribution of the number of individuals in each of the states after 4 generations (4 steps). b) What is the probability that after 4 generations there will be more than 30 professionals, more than 40 skilled workers and fewer than 50 unskilled