Figure 1 shows a sample path of a Poisson process where the sequence of random variables, {S1, S2, ...}, show the arrival times and the sequence, {X1, X2, ...}, consists of the interarrival times that are i.i.d. exponential random variables with E[X;] = 1/A. %3D S, S, X Š, Š, Figure 1: A sample path of a Poisson process. nd A such that S = AX where X = [X1 X2 X3 X4]" and S = [S1 S2 S3 Sa]" . Find fs(s) using fx(x) and A (a) (b) Hint: You may take the determinant of A as 1. 2.

Q10. Customers arrive in a grocery store according to a Poisson process with rate 6 per hour. What is the probability that exactly one customer has arrived between 9:00 am and 9:20 am? (A) 2e-² (B) 6e-6 (C) 8e-6 (D) 1/3 (E) None of the above

Suppose the traffic light at a specific intersection changes to green at 4-minute intervals. If a car arrives at the traffic light at a random time, then X = waiting time in minutes, is uniformly distributed between 0 and 4.   Suppose a random sample of 35 cars are observed and their waiting time is recorded. What is the probability that the sample mean of the waiting times is less than 1.5 minutes?

Question 4: A popular website page experiences a loading error which occurs an average of 5 times per day. Let X represent the number of loading errors per day and assume X has a Poisson distribution. Suppose a random sample of thirty days is selected from the last 6 months. a) What is the probability of observing less than 3 loading errors on one of the days? b) What is the probability that the sample mean is observed to be more than 4 loading errors?