- #1
vampire2008
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1. Assume that earthquakes strike a certain region at random times that are exponentially distributed with mean 1 year. Volcanic eruptions take place at random times that are exponentially distributed with mean 2 years. What is the probability that there will be two earthquakes before the next volcanic eruption?
2. A certain geiger counter (an instrument that detects individual radioactive decay events) has not been well maintained, and hence after each one it detects, it will not detect another (no matter how many occur) until it has gone a full two seconds with no decays. (for example, if it detects one event at 3 seconds, and this is followed by events at 3.5, 4.5, 6 and 9 seconds, the counter will not detect those at 3.5, 4.5 and 6 seconds.) Assume decay events occur according to a Poisson process with rate 0.5. Find the expected value of the time between events detected by the counter.
3. Briefly explain why it would or would not be a good idea to model the births of humans on Earth over the next year as a Poisson process.
4. A poor soul who has the title of Core Math Director receives emails at exponentially distributed time intervals, with rate 1 per minute. the times required to respond to these are independent, exponentially distributed with rate 2 per minute. However, when the list of unanswered messages gets above 10, his stress level rises and his efficiency goes down, reducing the rate to 1 per minute. Set up a birth/death process that models the email queue for this beleaguered public servant. Specifically, describe the states, and list the birth rate λi and the death rate μi in each state.
5.Customers arrive at a certain restaurant according to a Poisson process with rate 3 per hour. a small percentage of the customers are actually undercover health inspectors (as well as being customers). they come to the restaurant according to a Poisson process with rate 0.01 per hour. if no customer has entered for 1/3 of an hour, what is the probability that the next customer is a health inspector?
2. A certain geiger counter (an instrument that detects individual radioactive decay events) has not been well maintained, and hence after each one it detects, it will not detect another (no matter how many occur) until it has gone a full two seconds with no decays. (for example, if it detects one event at 3 seconds, and this is followed by events at 3.5, 4.5, 6 and 9 seconds, the counter will not detect those at 3.5, 4.5 and 6 seconds.) Assume decay events occur according to a Poisson process with rate 0.5. Find the expected value of the time between events detected by the counter.
3. Briefly explain why it would or would not be a good idea to model the births of humans on Earth over the next year as a Poisson process.
4. A poor soul who has the title of Core Math Director receives emails at exponentially distributed time intervals, with rate 1 per minute. the times required to respond to these are independent, exponentially distributed with rate 2 per minute. However, when the list of unanswered messages gets above 10, his stress level rises and his efficiency goes down, reducing the rate to 1 per minute. Set up a birth/death process that models the email queue for this beleaguered public servant. Specifically, describe the states, and list the birth rate λi and the death rate μi in each state.
5.Customers arrive at a certain restaurant according to a Poisson process with rate 3 per hour. a small percentage of the customers are actually undercover health inspectors (as well as being customers). they come to the restaurant according to a Poisson process with rate 0.01 per hour. if no customer has entered for 1/3 of an hour, what is the probability that the next customer is a health inspector?