A Poisson process is a stochastic process that occurs over a certain time period. Within this time period, the Poisson process counts the occurrences of events. The Poisson process is a continuous time process and a type of Markov process. An example of a Poisson process can be thought of as a piece of Uranium slowly experiencing radioactive decay. The radioactive particles are emitted in accordance to a function of the remaining number of particles; emission rate declines as number of particles left declines. This situation is consistent with all the properties of a Poisson process. At time zero, there are no particles emitted. The intervals between emitted particles are independently distributed from one another. No counted occurrences are simultaneous. The probability of the particles emitted at any given interval is dependent only on the time interval. Lastly the probability distribution becomes a Poisson distribution. Poisson processes can be classified into a few types. These are homogeneous, non-homogeneous, spatial, and space-time.

# Poisson Processes

### How likely is it that exactly 1 customer will arrive in any 5 minutes? If there are no additional customers waiting (or walking up to the counter) when a particular customer begins her transaction, what is the probability that no one will be waiting in line when her transaction is finished?

People arrive a a particular sales counter at the rate of 6 in any 15 minute period and are served on a "first come first served" basis. Arrivals occur randomly throughout the hour and the arrival rate is the same during the entire business day. It takes 5 minutes to service a single customer. How likely is it that exactly 1

### Stochastic Process, Random Variables, Discrete Random Variables, and Probability Distributions

Please do only 1, 2, 6, 7, 9, 10, 11. 1. Define (a) stochastic process; (b) random variable; (c) discrete random variable; and (d) probability distribution. 2. Without using formulas, explain the meaning of (a) expected value of a random variable; (b) actuarial fairness; and (c) variance of a random variable. 6. (a) Wha

### Stochastic Models

Spare Parts for an Alarm System. Consider an alarm system that operates continuously. The alarm system contains a minicomputer that is critical for the alarm system to function properly. Accordingly, a large number of spare minicomputers are maintained, so that the minicomputer can be instantaneously replaced whenever it fail

### Poisson Process and Probability

Assume the accidents occur at random according to a Poisson process a rate of one every two days. a) What is the probability that exactly two accidents will occur in a particular two-day interval? b)What is the probability that one will have to wait at least two days to observe the next two accidents.

### Queueing Systems

This exam covers customer queuing systems including arrival rates, Poisson processes, probability density functions, steady states, operator utilisation, machine efficiency, and exponentially distributed service times.

### 4079-stats

We have two independent Poisson processes, Z1(t) and Z2(t), with respective parameters r and s. They are running parallel. If Y(t) is the process from the overlay of Z1(t) and Z2(t), show that Y(t) is also a Poisson process. What is the rate of Y(t)?

### A certain piece of machinery is known to fail according to a Poisson process. a) In a series of tests, the piece was let operate till failure, repaired immediately, and let operate till next failure and so on for 3 months. The total number failures observed were 3. If the intent of the test was to determine , was the test run for too long, too short or just about the appropriate length of time ?b) To minimize unscheduled shutdowns, the piece of machinery is to be inspected and maintained on a regular interval of X days. What X should you select to have 90% confidence that you will not see failures during operation?c) Based on the results from (b), estimate the likelihood of finding no failure in 2 consecutive time periods.d) Based on your results from (a) assign a probability distribution to  and update it by taking to account that you found no unscheduled shutdowns, under the maintenance schedule you found in (b) for 3 months.

A certain piece of machinery is known to fail according to a Poisson process. a) In a series of tests, the piece was let operate till failure, repaired immediately, and let operate till next failure and so on for 3 months. The total number failures observed were 3. If the intent of the test was to determine , was th