11. Cats arrive at the vets as a Poisson process with rate 5 per hour. Dogs arri
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Question
11. Cats arrive at the vets as a Poisson process with rate 5 per hour. Dogs arrive there as an independent Poisson process with rate 10 per hour. An arriving cat is sick with probability 1/2 and otherwise has fleas. An arriving dog is sick with probability 1/3 and otherwise has fleas. What is the expected number of animals (cats and dogs) that will 1 arrive between 12:00pm and 6:00pm? What is the expected number of sick cats that will arrive during a 2 hour period? What is the expected number of animals with fleas that will arrive during a 3 hour period? Given that 20 animals arrived between 10:00am and 11:00am, what is the expected number of dogs who arrived, and what is the expected number of flea-ridden dogs who arrived? Given that 20 animals arrived between 10:00am and 11:00am, what is the probability that none of them were flea-ridden cats, and what is the probability that exactly 3 of them were sick cats? Given that 20 animals arrived between 10:00am and 11:00am, what is the expected amount of time (in hours) between the 4th arrival and the 6th arrival, and what is the expected time of the first arrival?
Thinking Question: Let X be the amount of time (in hours) from 9:00am until the first arrival of a sick dog. It is very easy to find E[X]. The think question is this: Suppose that exactly 3 animals arrived between 9:00am and 9:20am. Let Y = X if X 1 3 and Y = 1 3 otherwise. That is, Y is the time until the first sick dog’s arrival if any arrived during those 20 minutes, and is 1/3 of an hour (i.e. 20 minutes) if not. Find E[Y ] (given that exactly 3 animals arrived).
Explanation / Answer
11. Cats arrive at the vets as a Poisson process with rate 5 per hour. Dogs arrive there as an independent Poisson process with rate 10 per hour. An arriving cat is sick with probability 1/2 and otherwise has fleas. An arriving dog is sick with probability 1/3 and otherwise has fleas. What is the expected number of animals (cats and dogs) that will 1 arrive between 12:00pm and 6:00pm? What is the expected number of sick cats that will arrive during a 2 hour period? What is the expected number of animals with fleas that will arrive during a 3 hour period? Given that 20 animals arrived between 10:00am and 11:00am, what is the expected number of dogs who arrived, and what is the expected number of flea-ridden dogs who arrived? Given that 20 animals arrived between 10:00am and 11:00am, what is the probability that none of them were flea-ridden cats, and what is the probability that exactly 3 of them were sick cats? Given that 20 animals arrived between 10:00am and 11:00am, what is the expected amount of time (in hours) between the 4th arrival and the 6th arrival, and what is the expected time of the first arrival?
Thinking Question: Let X be the amount of time (in hours) from 9:00am until the first arrival of a sick dog. It is very easy to find E[X]. The think question is this: Suppose that exactly 3 animals arrived between 9:00am and 9:20am. Let Y = X if X 1 3 and Y = 1 3 otherwise. That is, Y is the time until the first sick dog’s arrival if any arrived during those 20 minutes, and is 1/3 of an hour (i.e. 20 minutes) if not. Find E[Y ] (given that exactly 3 animals arrived).
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