Problem 4. Suppose that in any given hour of the day, the number of clients entering a post-office is a random variable N, with Poisson distribution of parameter 4. In other words, -1 P(N = n) = n=0,1,2,.... nal Suppose that the post office has two service counters, denoted A and B. For each client i = 1,...,N, let y be the random variable which is equal to 1 if and only if client i chooses to queue at counter A 2 and which equals 0, if(6)be chooses to queue at counter B. Similarly, for each client i = 1,..., N, let Z be the random variable which equals 1 if and only if client i abandons the queue before being served, independently of the chosen queue, and decides to come back at a later time. We will further suppose that each Y and Z, besides being mutually independent, are jointly independent of N as well. Finally, suppose that there exist p.9 (0,1) such that, for all i, P(Y = 1) =p and P(Z = 1) = 9. In the sequel, it will be useful to observe that, for each 1, the variable Xi :=Yi(l-Zi) is a Bernoulli random variable which takes the value 1 if and only if client i ends up being served and that (she is served at counter A 1. Compute the probability that a given client ends being served and that (she is be served at counter B 2. Compute the expected number of clients which will be served in any given hour. 3. Let Mg be the number of clients which end up being served at counter B, in any given hour. Compute P(Mg = m), m = 0,1,2,... Deduce E(Mg) and var(Mg). 4. Define the random variable W := min{N, S), where S :=min{i > 1|YZ1 = 1) What quantity does this variable describe ? Compute E(W). Problem 4. Suppose that in any given hour of the day, the number of clients entering a post-office is a random variable N, with Poisson distribution of parameter 4. In other words, -1 P(N = n) = n=0,1,2,.... nal Suppose that the post office has two service counters, denoted A and B. For each client i = 1,...,N, let y be the random variable which is equal to 1 if and only if client i chooses to queue at counter A 2 and which equals 0, if(6)be chooses to queue at counter B. Similarly, for each client i = 1,..., N, let Z be the random variable which equals 1 if and only if client i abandons the queue before being served, independently of the chosen queue, and decides to come back at a later time. We will further suppose that each Y and Z, besides being mutually independent, are jointly independent of N as well. Finally, suppose that there exist p.9 (0,1) such that, for all i, P(Y = 1) =p and P(Z = 1) = 9. In the sequel, it will be useful to observe that, for each 1, the variable Xi :=Yi(l-Zi) is a Bernoulli random variable which takes the value 1 if and only if client i ends up being served and that (she is served at counter A 1. Compute the probability that a given client ends being served and that (she is be served at counter B 2. Compute the expected number of clients which will be served in any given hour. 3. Let Mg be the number of clients which end up being served at counter B, in any given hour. Compute P(Mg = m), m = 0,1,2,... Deduce E(Mg) and var(Mg). 4. Define the random variable W := min{N, S), where S :=min{i > 1|YZ1 = 1) What quantity does this variable describe ? Compute E(W)
