Customers arrive at a soft drink dispensing machine according to a Poisson process with

rate ? per hour. Let N(t) be the number of custoer arrivals up to time t, with hour as the

unit. There are two types of soft drinks, type A and B, stored in the machine. Suppose

that each time a customer deposits money, the machine dispenses one soft drink A

with

probability p1, or one soft drink B with probability p2. We have p1 + p2 = 1, p1

> 0,

p2 > 0. Let X(t) be the number of type A soft drinks dispensed up to time t; and

Y (t)

be the number of type B soft drinks dispensed up to time t.

2. Customers arrive at a soft drink dispensing machine according to a Poisson process with rate /\\ per hour. Let N (t) be the number of custoer arrivals up to time t, with hour as the unit. There are two types of soft drinks, type A and B, stored in the machine. Suppose that each time a customer deposits money, the machine dispenses one soft drink A with probability 391, or one soft drink B with probability p2. We have 301 + 392 = 1, p1 > 0, p2 > 0. Let X (t) be the number of type A soft drinks dispensed up to time t; and Y(t) be the number of type B soft drinks dispensed up to time t. (a) (6) Specify the pdf of the arrival time of the rst customer. (b) (6) Suppose 2 customers arrived in the rst hour. What is the probability that both arrived in the rst 20 minutes? (c) (7) Find P{N(3) : m and N(4) : n|N(1) : k}, n 2 m 2 k 2 0. 1 (d) (5) What is the probability that 2 type B soft drinks have been dispensed up to time t given that N(t) : 10? (e) (5) Given that one type B soft drink has been dispensed up to time t, what is the expected number of type A soft drinks that have been dispensed up to time t? (f) (5) Suppose that one customer arrived during the rst one hour, i.e., N (1) : 1. Find the joint probability mass function (pmf) of X(U.5) and Y(0.5), i.e., the numbers of type A and type B soft drinks dispensed in the rst half an hour. (g) (5) Suppose X (t) = 2, nd the conditional distribution of N(t), that is, specify the probability mass function for N (t)