Question

Parishioners arrive at church on Sunday morning according to a Poisson process at rate 2/minute starting at 9:00am until 11:00am. Each parishioner wears a hat with probability 1/3, independent of other parishioners, and brings an umbrella with probability 1/4, independent of whether she wears a hat and independent of other parishioners. The cloakroom has umbrella stands and baskets for hats.

The sermon begins at 11:00am. Once the sermon begins, each parishioner falls asleep after an exponentially distributed amount of time with mean 12 minutes, independent of the others. The sermon ends when all parishioners are asleep.

i. If exactly 300 parishioners arrive by 11:00am, what is the expected time at which the sermon ends?   

ii. What is the expected number of sleeping parishioners (in the church) at 11:30am?

iii. What is the probability that at least one parishioner is (in church and) awake at noon? What is the probability that exactly three parishioners are (in church and) awake at 12:12pm?

iv. All of the parishioners who brought umbrellas have colds; they sniffle and sneeze until they fall asleep. The other parishioners are healthy but sleep-deprived; once the sermon begins they each fall asleep after 10 minutes. However, whenever someone sniffles or sneezes, it alarms every healthy parishioner, each of whom won’t sleep for another 10 minutes. What is the probability that everyone is asleep by 12:34pm?

Answer 1

Answer: I Since hats amine at rate 2/3 per minute for 120 minutes. (619-Ilam). then mean(u )- 2/3 X120 = 80min. So, since no. of hats is poission with man 80 min Pysko) ple = 300) = é80(80 3900 500! - In poisson disto P&=x)=). ii) Assume that Each parishioner decide to church to ogleep at 11:300m with. Probability 1-2 30/R-1-22-s then slep inclined parishioners arrive as at date a cl-e2.s) per min for 120 min then to Expected number in 240 Cirezer) 811) parishioner & decide to asleep at 11:30 am 12112.pm assume). ie 42min then will sleep after well anale Parishioners arrive rate zire2s), per min for 120min. then Expected number is 200 Cre2s) with 45minic N 285 mun Cizo eus) 1. then 330 C1-es), then, the parishioner awake at asahe at 330(1-625) - 240 C1-825) = 330 - 330425240 +240025 = 90-90 e 25 = 861-825)