Question: Customers arrive at a 24-hour store as a Poisson process. Men arrive as a Poisson process rate 3 per hour, women at a rate 9/hour,

Customers arrive at a 24-hour store as a Poisson process. Men arrive as a Poisson process rate 3 per hour, women at a rate 9/hour, so the total process has rate 12 per hour, and each arriving customer is independently a man with probability 1/4, and a woman with probability 3/4. 1.

Door security agent Fred comes on duty at 7:00 a.m., and observes arriving customers.

(a) What is the probability he observes a total of 24 customers arriving by 9:00 a.m.? What are the mean and variance?

(b) What is the probability he observes 7 men and 20 women arrriving between 9:00 a.m and 11:00 a.m., but only a total of 8 customers between 11:00a.m. and 12:00 noon

2. At noon Fred takes a short 10-minute break.

(a) What is the probability that no customers arrive during his break?

(b) What is the pdf of the arrival interval (after Fred returns at 12:10 pm) until the next woman customer arrives? What is the mean and standard deviation

(c) What is the pdf of the time interval between the arrival of the last customer before Fred came on duty at 7:00 a.m. and 7:00 a.m?

3. (a) Given one male customer arrived between 6:45 a.m. and 7:00 a.m., what is the pdf of the time interval before 7:00 a.m. of the arrival of this customer?

(b) What is the mean?, what is the variance?

4. Dave comes on duty at 2:00 pm, after Fred completes his shift.

(a) What is the pmf of the number of customers that arrive up to and including the first male customer (after 2:00 pm)?

(b) What is the probability that the first male customer is the 5th customer, and that the second male customer is the 8 th customer?

(c) Given that the second male customer is the 8th customer, what is the probability that the first male customer was the fifth customer?

5. Dave continues his shift unti 6:00pm

(a) Conditional of the fact that Dave observes a total of 30 customers between 4:00pm and 6:00 pm, what is the pmf of the number of women customers between 4:00pm and 6:00 pm? What are the mean and variance?

(b) What are the mean and variance of the number of women customers between 6:00 pm and 8:00pm? (All door sta now o-duty)

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