Poisson+distribution

=Poisson Distribution - The Porsche (P for Poisson and Porche)= The Poisson Distribution was developed by the French mathematician Simeon Denis Poisson in 1837. The Poisson random variable satisfies the following conditions: Apart from disjoint time intervals, the Poisson random variable also applies to disjoint regions of space.
 * 1) The number of successes in two disjoint time intervals is independent.
 * 2) The probability of a success during a small time interval is proportional to the entire length of the time interval.

Attributes of a Poisson Experiment
A **Poisson experiment** is a statistical experiment that has the following properties: Note that the specified region could take many forms. For instance, it could be a length, an area, a volume, a period of time, etc.
 * The experiment results in outcomes that can be classified as successes or failures.
 * The average number of successes (μ) that occurs in a specified region is known.
 * The probability that a success will occur is proportional to the size of the region.
 * The probability that a success will occur in an extremely small region is virtually zero.

Applications

 * the number of deaths by horse kicking in the Prussian army (first application)
 * birth defects and genetic mutations
 * rare diseases (like Leukemia, but not AIDS because it is infectious and so not independent) - especially in legal cases
 * car accidents
 * traffic flow and ideal gap distance
 * number of typing errors on a page
 * hairs found in McDonald's hamburgers
 * spread of an endangered animal in Africa
 * failure of a machine in one month

Notation
The following notation is helpful, when we talk about the Poisson distribution.
 * //e//: A constant equal to approximately 2.71828. (Actually, //e// is the base of the natural logarithm system.)
 * μ: The mean number of successes that occur in a specified region.
 * //x//: The actual number of successes that occur in a specified region.
 * P(//x//; μ): The **Poisson probability** that __exactly__ //x// successes occur in a Poisson experiment, when the mean number of successes is μ.

Poisson Distribution
A **Poisson random variable** is the number of successes that result from a Poisson experiment. The probability distribution of a Poisson random variable is called a **Poisson distribution**. Given the mean number of successes (μ) that occur in a specified region, we can compute the Poisson probability based on the following formula: **Poisson Formula.** Suppose we conduct a Poisson experiment, in which the average number of successes within a given region is μ. Then, the Poisson probability is: P(//x//; μ) = (e-μ) (μx) / x! where //x// is the actual number of successes that result from the experiment, and //e// is approximately equal to 2.71828. The Poisson distribution has the following properties:
 * The mean of the distribution is equal to μ.
 * The variance is also equal to μ.


 * Example 1**

The average number of homes sold by the Acme Realty company is 2 homes per day. What is the probability that exactly 3 homes will be sold tomorrow? //Solution:// This is a Poisson experiment in which we know the following: We plug these values into the Poisson formula as follows: P(//x//; μ) = (e-μ) (μx) / x! P(3; 2) = (2.71828-2) (23) / 3! P(3; 2) = (0.13534) (8) / 6 P(3; 2) = 0.180
 * μ = 2; since 2 homes are sold per day, on average.
 * x = 3; since we want to find the likelihood that 3 homes will be sold tomorrow.
 * e = 2.71828; since //e// is a constant equal to approximately 2.71828.

Thus, the probability of selling 3 homes tomorrow is 0.180.