Poisson process

Examples of Poisson process in the following topics:

• The Exponential Distribution

  • The exponential distribution is a family of continuous probability distributions that describe the time between events in a Poisson process.
  • It describes the time between events in a Poisson process (the process in which events occur continuously and independently at a constant average rate).
  • The exponential distribution describes the time for a continuous process to change state.
  • The length of a process that can be thought of as a sequence of several independent tasks is better modeled by a variable following the Erlang distribution (which is the distribution of the sum of several independent exponentially distributed variables).

• The Poisson Random Variable

  • The Poisson distribution is a discrete probability distribution.
  • Given only the average rate for a certain period of observation (i.e., pieces of mail per day, phonecalls per hour, etc.), and assuming that the process that produces the event flow is essentially random, the Poisson distribution specifies how likely it is that the count will be 3, 5, 10, or any other number during one period of observation.
  • The Poisson random variable, then, is the number of successes that result from a Poisson experiment, and the probability distribution of a Poisson random variable is called a Poisson distribution.
  • P(x; μ): the Poisson probability that exactly x successes occur in a Poisson experiment, when the mean number of successes is μ; and
  • This is a Poisson experiment in which we know the following:

• Poisson Distribution

  • The Poisson distribution can be used to calculate the probabilities of various numbers of "successes" based on the mean number of successes.
  • In order to apply the Poisson distribution, the various events must be independent.
  • This problem can be solved using the following formula based on the Poisson distribution:
  • The mean of the Poisson distribution is μ.

• Notation for Poisson: P = Poisson Probability Distribution Function X ~ P(u)

  • Read this as "X is a random variable with a Poisson distribution. " The parameter is µ (or λ). µ (or λ) = the mean for the interval of interest.
  • The Poisson parameter list is (µ for the interval of interest, number).

• Common Discrete Probability Distribution Functions

  • Some of the more common discrete probability functions are binomial, geometric, hypergeometric, and Poisson.
  • Most elementary courses do not cover the geometric, hypergeometric, and Poisson.

• Poisson distribution

  • The Poisson distribution is often useful for estimating the number of rare events in a large population over a unit of time.
  • The Poisson distribution helps us describe the number of such events that will occur in a short unit of time for a fixed population if the individuals within the population are independent.

• Practice 3: Poisson Distribution

• Poisson

  • The Poisson gives the probability of a number of events occurring in a fixed interval of time or space if these events happen with a known average rate and independently of the time since the last event.
  • The Poisson may be used to approximate the binomial if the probability of success is "small" (such as 0.01) and the number of trials is "large" (such as 1000).
  • This is a Poisson problem because you are interested in knowing the number of times the news reporter says "uh" during a broadcast.

• Student Learning Outcomes

• Summary of Functions

  • The mean µ is typically given. ( λ is often used as the mean instead of µ. ) When the Poisson is used to approximate the binomial, we use the binomial mean µ = np. n is the binomial number of trials. p = the probability of a success for each trial.
  • If n is large enough and p is small enough then the Poisson approximates the binomial very well.