Introduction to Probability Function (Part3)

3. Poisson Distribution 

In probability theory and statistics, the Poisson distribution , named after French mathematician Siméon Denis Poisson, is 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.The Poisson distribution can also be used for the number of events in other specified intervals such as distance, area or volume.

In the picture above are simultaneously portrayed several Poisson distributions. Where the rate of occurrence of some event, r (in this chart called lambda or l) is small, the range of likely possibilities will lie near the zero line. Meaning that when the rate r is small, zero is a very likely number to get. As the rate becomes higher (as the occurrence of the thing we are watching becomes commoner), the center of the curve moves toward the right,This is how the Poisson world looks graphically.

A discrete random variable X  is said to have a Poisson distribution with parameter λ > 0, if, for k = 0, 1, 2, …, the probability mass function of X  is given by:

\!f(k; \lambda)= \Pr(X = k)= \frac{\lambda^k e^{-\lambda}}{k!},

where

  • e is Euler’s number (e = 2.71828…)
  • k! is the factorial of k.

The positive real number λ is equal to the expected value of X and also to its variance

\lambda=\operatorname{E}(X)=\operatorname{Var}(X).
Application

The Poisson distribution applies when: (1) the event is something that can be counted in whole

numbers; (2) occurrences are independent, so that one occurrence neither diminishes nor increases the chance of another; (3) the average frequency of occurrence for the time period in question is known; and (4) it is possible to count how many events have occurred, such as the number of times a firefly lights up in my garden in a given 5 seconds, some evening, but meaningless to ask how many such events have not occurred. This last point sums up the contrast with the Binomial situation, where the probability of each of two mutually exclusive events (p and q) is known. The Poisson Distribution, so to speak, is the Binomial Distribution Without Q. In those circumstances, and they are surprisingly common, the Poisson Distribution gives the expected frequency profile for events. It may be used in reverse, to test whether a given data set was generated by a random process. If the data fit the Poisson Expectation closely, then there is no strong reason to believe that something other than random occurrence is at work. On the other hand, if the data are lumpy, we look for what might be causing the lump.

Summary

  • The Poisson distribution deals with mutually independent events, occurring at a known and constant rate λ  per unit (of time or space), and observed over a certain unit of time or space.
  • The probability of k occurrences in that unit can be calculated from
\!f(k; \lambda)= \Pr(X = k)= \frac{\lambda^k e^{-\lambda}}{k!},
  • The rate λ is the expected or most likely outcome (for whole number r greater than 1, the outcome corresponding to λ -1 is equally likely).
  • The frequency profile of Poisson outcomes for a givenλ is not symmetrical; it is skewed more or less toward the high end.
  • For Binomial situations with p < 0.1 and reasonably many trials, the Poisson Distribution can acceptably mimic the Binomial Distribution, and is easier to calculate.

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