Poisson distribution

(noun)

A discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time and/or space, if these events occur with a known average rate and independently of the time since the last event.

Related Terms

Examples of Poisson distribution in the following topics:

  • 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.

  • 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

  • The Poisson Random Variable

    • The Poisson distribution is a discrete probability distribution.
    • The Poisson distribution can also be used for the number of events in other specified intervals such as distance, area, or volume.
    • 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.
    • Applications of the Poisson distribution can be found in many fields related to counting:
    • Examples of events that may be modelled as a Poisson distribution include:

  • 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).

  • Student Learning Outcomes

  • Additional Properties of the Binomial Distribution

    • In this section, we'll look at the median, mode, and covariance of the binomial distribution.
    • The binomial distribution is a special case of the Poisson binomial distribution, which is a sum of n independent non-identical Bernoulli trials Bern(pi).
    • If X has the Poisson binomial distribution with p1=…=pn=pp1=\ldots =pn=p then ∼B(n,p)\sim B(n, p).
    • This formula is for calculating the mode of a binomial distribution.
    • This summarizes how to find the mode of a binomial distribution.

  • 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.
    • A probability distribution function is a pattern.
    • These distributions are tools to make solving probability problems easier.
    • Each distribution has its own special characteristics.

  • 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.

  • The Exponential Distribution

    • The exponential distribution is a family of continuous probability distributions that describe the time between events in a Poisson process.
    • The exponential distribution is a family of continuous probability distributions.
    • 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 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).
    • Reliability engineering also makes extensive use of the exponential distribution.