Poisson distribution

Discrete variable

A quantitative variable that has either a finite number of possible values or a countable number of possible values

Poisson distribution mean

Another important point is that λt, the average number in t segments, is not necessarily the number we will see if we observe the process for t segments. We might expect an average of 20 people to arrive at a checkout stand in any given hour, but we do not expect to find exactly that number arriving every hour. The actual arrivals will form a distribution with an expected value, or mean equal to λt. So for the Poisson distribution, E[x] =µx = λt Once λ and t have been specified, the probability for any discrete value in the Poisson distribution can be found using using equation P(x) =(λt)^x * e-λt / x! t = number of segments of interest x = number of successes in t segments λ = expected number of successes in one segment e = base of the natural logarithm system (2.781828)

Reason for Poisson Distribution

In a binomial distribution we must be able to count the number of successes and the number of failures. So in many situations we can count the number of successes but not the number of failures. Ex: Hospital supplying emergency medical services in Los Angeles. It could easily count the number of emergencies its units respond to in one hour, but how could it determine how many calls it did not receive. So in cases where the number of possible outcomes of (successes + failures) is difficult or impossible to determine a binomial distribution cannot be applied. This is where you might be able to use the Poisson distribution.

Poisson Distribution

The Poisson distribution describes a process that extends over space, time, or any well-defined interval or unit of inspection in which the outcomes of interest occur at random and the number of outcomes that occur in any given interval are counted. The poison distribution, is used when the total number of possible outcomes cannot be determined.

factorial

The factorial function (symbol: !) says to multiply all whole numbers from our chosen number down to 1. Examples: 4! = 4 × 3 × 2 × 1 = 24.

Poisson Probability distribution assumptions

We can use the Poisson probability distribution to answer these questions if we make the following assumptions: 1. We know (λ), the average number of successes in one segment. For example, we know that there is an average of 8 emergency calls per hour (λ = 8) or an average of 15 potholes per mile of freeway (λ = 15). 2. The probability of x successes in a segment is the same for all segments of the same size. For example, the probability distribution of emergency calls is the same for any one-hour period of time at the hospital. 3. What happens in one segment has no influence on any nonoverlapping segment. For example the number of calls between the number of calls arriving between 9:30 PM and 10:30 PM has no influence on the number of calls between 11:00PM and 12:00 PM midnight. 4. We imagine dividing time or space into tiny subsegments. Then the change of more than one success in a subsegment is negligible and the chance of exactly one success in a tiny subsegment of length t is (λt) For example, the chance of two emergency calls in the same second is essentially 0, and if λ = 8 calls per hour, the chance of a call in any given second (8)(1/3,600) = 0.0022. Once lambda has been determined, we can calculate the average occurrence rate for any number of segments (t). This is at λ(t). Note that λ and t must be compatible units. If we have λ = 20 arrivals per hour, the segments must be in hours or fractional parts of an hour. That is, if we have λ= 20 per hour and we wish to work with half-hour periods the segment would be t = 1/2hour, not t = 30 minutes.

Binomial Distribution

a frequency distribution of the possible number of successful outcomes in a given number of trials in each of which there is the same probability of success. Describes processes whose trials have two possible outcomes. A distribution that gives the probability of x successes in n trials in a process that meets the following conditions: 1 A trial has only two possible outcomes; a success or failure. 2. There is a fixed number, n, of identical trials 3. The trials of the experiment are independent of each other. This means that if one outcome is a success, this does not influence the change of another outcome being a success. 4. The process must be consistent in generating successes and failures. That is, the probability p, associated with a success remains constant from trial to trial 5. if p represents the probability of a success, then (1 - p) = q is the probability of a failure.

continuous variable

a quantitative variable that has an infinite number of possible values that are not countable

Poisson Distribution description

describes a process that extends over time, space, or any well-defined unit of inspection. Ex: The outcomes of interest, such as emergency calls or potholes occur at random, and we count the number of outcomes that occur in a given segment of time or space. We might count the number of emergency calls in a one hour period, or the number of potholes in a two-mile stretch of freeway. As we did with the binomial distribution, we will call these outcomes successes even though (like potholes) they might be undesirable. The possible counts are the integers 0,1,2,3, etc... and we would like to know the probability of each of these values. For Example, what is the chance of getting exactly four emergency calls in a particular hour? What is the chance that a chosen two mile stretch of freeway will contain zero pot holes.