5. Messages arrive at a telegraph office as a Poisson process with mean rate of 3 messages per hour. What is the i) ii) Probability that no messages arrive during the morning hours 8:00 A.M. to noon? Distribution of the time at which the first afternoon message arrives? 6. Suppose that customers arrive at a facility according to a Poisson process having rate = 2. Let x(t) be the number of customers that have arrived up to time t. Determine the following probabilities and conditional probabilities: (i) Pr[X(1)=2] (ii) Pr[X(1)=2 and X(3)=6] (iii) Pr[X(1)=2/X(3) = 6] (iv) Pr[X(3)=6/X(1)=2] 7. Let {X(t); t 0} be a Poisson process having rate parameter = 3. Determine the numerical values to two decimal places for the following probabilities: a) Pr[X(1) 2] b) Pr[X(1) 3] c) Pr[X(1)=2 and X(2) >3] d) Pr[X(1)2/X(1) 1] e) E[X(2)] f) E[X(1)X(3)] 8. A radioactive source emits particles according to a Poisson process of rate =3 particles per minute. What is the probability that the i) First particle appears after three minutes ii) First particle appears some time after three minutes but before five minutes? iii) Exactly one particle is emitted in the interval from three to five minutes?